Prosecutors and Hunter Biden’s team were still trying to negotiate a plea deal this week.

But a bitter debate between prosecutors and the defense team over the government’s obligation to stick by the terms of the original agreement scuttled those efforts for good. It is unusual for the government to renege on an agreement that one of its prosecutors has signed, but there is nothing usual about Mr. Biden’s case.

Mr. Weiss eventually opted for a second track he had also been considering — asking Attorney General Merrick B. Garland to name him special counsel.

One of the key sticking points in the talks, people familiar with the situation said, was whether the original agreement was still technically in place.

Mr. Weiss took the position that although a prosecutor working for him had signed the deal, it was only a draft that could be subject to revision, or even abandoned depending on the circumstances. Mr. Biden’s lawyers strongly objected and demanded that the government abide by the previous agreement.

Mr. Weiss proposed a new one. His offer preserved the core of the deal hammered out over the past several months — no prison time in exchange for a guilty plea on tax misdemeanors and enrollment in a diversion program for nonviolent gun offenders — but without the broad immunity guarantee for Mr. Biden, the people familiar with the situation said.

It was a deal breaker. Immunity had been one of the most important pillars of the deal for Mr. Biden, and it became clear to the government that Mr. Biden’s lawyers would never agree to that, the people said. Because of this, prosecutors could see that they would have to take a different tack, raising the possibility that Mr. Biden might now face a fresh criminal indictment and a possible trial.

It is not clear when Mr. Weiss began discussing his possible appointment as a special counsel with Mr. Garland, although he appears to have begun considering the possibility while still negotiating with Mr. Biden’s legal team.

Adam Entous contributed reporting.

And Nikki Haley, the former U.S. ambassador to the United Nations, spoke disdainfully of the new title and power for Mr. Weiss.

“I don’t think the American people trust the Department of Justice or anything this is going to do,” Ms. Haley said in an appearance on Fox News. “I think this was meant to be a distraction.”

At the same time, she called it a “response to the pressure that the Biden family is feeling” and called on House Republicans who have been investigating the Bidens “to keep their foot on the gas.” So far, the investigations have found no hard evidence that President Biden used his influence while vice president to benefit his son’s business deals.

Gov. Doug Burgum of North Dakota also said he doubted the independence of Mr. Weiss. In an interview while twirling around on the Iowa State Fair’s Ferris wheel, he called the move “too little, too late” and said few Republicans would view the step as a serious development, given Mr. Weiss’s role in offering Hunter Biden a plea deal in the case. That plea deal has fallen apart.

Mr. Weiss is a federal prosecutor in Delaware who was originally appointed by former President Donald J. Trump. He was left in his position by President Biden to continue the Hunter Biden inquiry to avoid the appearance that the president would seek special treatment for his son.

In a statement attributed to a spokesperson, Mr. Trump, who is being investigated by the special counsel Jack Smith, claimed without evidence that the Department of Justice has protected President Biden, Hunter Biden and other family members “for decades.” The statement cast doubt on Mr. Weiss’s independence and criticized him for not already bringing “proper charges after a four-year investigation” of Hunter Biden.

Mr. Smith has brought two indictments against Mr. Trump.

Not all of the candidates were disdainful of the appointment of the special counsel, which Republicans have urged for some time. Vivek Ramaswamy, who said last month that a special counsel was warranted, called the appointment of Mr. Weiss “good” on X, the site formerly known as Twitter. “Now let’s see if it’s more than a fig leaf,” he added.

Former Vice President Mike Pence, who was flipping cuts of pork at the Iowa State Fair on Friday, said he approved of the Department of Justice’s move to upgrade Mr. Weiss’s power.

“I think it’s about time that we saw the appointment of a special counsel to get to the bottom of not only what Hunter Biden was doing, but what the Biden family was doing,” Mr. Pence said. “The American people deserve answers, and I welcome the appointment.”

Anjali Huynh and Neil Vigdor contributed reporting from Iowa.

The attorney general’s move came against the backdrop of accusations by Republicans that Mr. Weiss had offered what they portrayed as a sweetheart plea bargain to the younger Mr. Biden because of political manipulations by Mr. Garland or by the White House. Functionally, the formalization of Mr. Weiss’s independence could serve as a shield against such accusations.

A special counsel is a prosecutor who wields the same powers as a U.S. attorney but is granted broader day-to-day independence from supervision. In making the announcement, Mr. Garland reminded the public that he had already said Mr. Weiss, who was appointed by President Trump, was operating outside the normal system of hierarchical oversight and control for the Hunter Biden case.

Discussing the decision to keep Mr. Weiss on to complete the Hunter Biden investigation in February 2021, after the Biden administration took office and most Trump political appointees resigned, Mr. Garland noted: “As I said before, Mr. Weiss would be permitted to continue his investigation, take any investigative steps he wanted, and make the decision whether to prosecute in any district.”

He added: “Mr. Weiss has told Congress that he has been granted ultimate authority over this matter, including the responsibility for deciding where, when, and whether to file charges and for making decisions necessary to preserve the integrity of any prosecution, consistent with federal law, the principles of federal prosecution, and departmental policies.”

In many ways, that description of Mr. Weiss’s previous authority does not sound notably different from the power he will now wield under the rules for special counsels, who also operate with day-to-day autonomy. In a statement on Friday, Chris Clark, a defense attorney for Hunter Biden, said the prosecutor’s new status did not appear to be a substantive change.

“This doesn’t change our understanding of Mr. Weiss’ authority over the five-year investigation into Mr. Biden,” he said. “For years, both Mr. Weiss and the department have assured us and the public that Mr. Weiss had more authority than a special counsel and full authority to negotiate a resolution of his investigation — which has been done.”

The differences between the rules governing U.S. attorneys and special counsels are less about law enforcement powers than they are about protections intended to bolster public confidence that they are operating free from political interference. For example, special counsels cannot be arbitrarily fired without a cause, like a finding of misconduct.

An attorney general may only block a special counsel from bringing a charge if the attorney general decides the step would be “so inappropriate or unwarranted under established departmental practices that it should not be pursued” — a high bar to meet. And at the end of the inquiry, Congress must be told about the decision, which creates a deterrent to abusing that power.

There is one notable requirement imposed upon special counsels: at the end of their investigations, they must write a report to the attorney general. While the rules envision that being a private document, the attorney general can choose to disclose it. Mr. Garland said he intended to release as much as possible of Mr. Weiss’s eventual report.

A factor in Mr. Garland’s decision not to name Mr. Weiss a special counsel from the start of the Biden administration, despite granting him broader independence, may be that the regulation envisions that special counsels will be people brought in from outside the Justice Department. Bestowing that status on a sitting U.S. attorney — Mr. Garland said Mr. Weiss would also continue in that role for Delaware — is unusual.

For example, Robert S. Mueller III, the special counsel appointed in 2017 to look into Russian interference in the 2016 election, was a retired F.B.I. director. Similarly, Jack Smith and Robert Hur, the special counsels investigating former President Donald Trump and President Biden, had previously been federal prosecutors, but neither was still working for the Justice Department when Mr. Garland named them to their current roles.

(It is not unprecedented, however. For example, Attorney General William P. Barr made John Durham, then the U.S. attorney for Connecticut, a special counsel to entrench him so he could continue to investigate the F.B.I.’s handling of the Russia inquiry even if Mr. Trump lost the election. Mr. Durham resigned as U.S. attorney in January 2021, but for several months he held both roles.)

Mr. Garland on Friday noted that Mr. Weiss had said in a letter to Congress last month that he had never requested special counsel status. But, Mr. Garland said, on Aug. 8, Mr. Weiss “advised me that in his judgment, his investigation has reached a stage at which he should continue his work as a special counsel, and he asked to be so appointed.”

Mr. Garland also said he had decided that taking that step now would be in the public interest. He did not say what had changed.

Republicans in Congress have stepped up their accusations recently that the investigation of Hunter Biden has been politicized, including taking testimony by an I.R.S. official, Gary Shapley, who claimed Mr. Weiss had sought special counsel status and had been turned down, an assertion Mr. Weiss denied. Granting him formal special counsel status could undercut such attacks.

But Mr. Weiss also unveiled a change to the investigation in court papers he filed shortly after Mr. Garland’s announcement: guilty plea negotiations had collapsed, he said, and that he now expected the tax charges against Hunter Biden — and possibly additional ones — to go to trial.

Mr. Weiss grew up in the Philadelphia suburbs, attending Cheltenham High School, whose graduates include Prime Minister Benjamin Netanyahu of Israel, the Hall of Fame slugger Reggie Jackson and the conservative commentator Mark Levin.

He graduated from Washington University in St. Louis. After completing law school at Widener University in Wilmington, Del., he worked two long stints as a prosecutor in the Delaware U.S. attorney’s office on an array of cases. Those included an investigation of a businessman who pleaded guilty to campaign finance violations after illegally bundling contributions for President Biden’s failed 2008 bid for the White House.

Mr. Weiss, graying and bespectacled now, once bore a striking resemblance to the singer Tony Orlando. In early 2018, when his colleagues celebrated him after his unanimous confirmation by the Senate, they did so against the backdrop of an old, blown-up photo from the 1970s of Mr. Weiss with long, center-parted hair.

He has been, until now, a noncontroversial figure, earning the support of both of Delaware’s Democratic senators when Mr. Trump, now a critic, nominated him in late 2017.

“David is a career prosecutor and dedicated public servant, longtime Delawarean and valued member of our law enforcement community,” Senator Chris Coons said in a statement at the time, according to The News Journal.

In announcing his nomination, Mr. Trump said in a statement that Mr. Weiss would “share the president’s vision for ‘Making America Safe Again.’”

People who know Mr. Weiss said that the Biden case, the best-known investigation he has overseen, is not what he believes will define his legacy.

In the late 1990s, while in private practice, Mr. Weiss began looking into the disappearance of Anne Marie Fahey, a secretary for Delaware’s governor — prodding his friends in the U.S. attorney’s office to join the local investigation, at the behest of Ms. Fahey’s family.

The trail eventually led to Thomas Capano, a former boyfriend of Ms. Fahey’s and a well-connected former Delaware deputy attorney general. He was convicted of murdering Ms. Fahey after their relationship ended and then dumping her body into the ocean.

“He would tell you that was his most important case,” said Thomas W. Ostrander, a lawyer who worked with Mr. Weiss for Ms. Fahey’s family. “We were in regular contact with them every day, for months and months, and David guided them through everything, because he was such an experienced trial lawyer and knew the pitfalls and challenges of putting together such a complex case.”

Mr. Biden has also pointed to his son’s sobriety as Hunter Biden recovers from a crack cocaine addiction: “My son, like a lot of people we know at home, had a drug problem. He’s overtaken it. He’s fixed it. He’s worked on it, and I’m proud of him,” Mr. Biden said during a 2020 presidential debate.

Hunter Biden’s addiction spiraled out of control after the 2015 death of his brother, Beau, and his dealings with foreign governments caught the attention of conservatives. The younger Mr. Biden’s choices have become grist for memes, conservative cable news panels and Republican fund-raising. In the White House, matters involving Hunter Biden are so sensitive that only the president’s most senior advisers talk to him about his son, according to people familiar with the arrangement.

Father and son remain extremely close. Hunter Biden, 53, is the president’s last surviving son and the last link to the car crash that killed the president’s first wife and infant daughter in December 1972.

When they are not together, they speak several times a day, and the president has always trusted his son’s political instincts. Mr. Biden has never treated his son like a political problem, choosing instead to keep him at his side, whether it’s at a state dinner or a ride on Marine One.

After Hunter Biden attended a recent State Dinner at the White House, in the wake of reaching a plea agreement on possible tax and gun charges, some Democrats privately complained about the appearance, though few were willing to say it publicly.

Under the previous agreement, Mr. Biden would have pleaded guilty to two tax misdemeanors and averted prosecution on a gun charge by enrolling in a two-year diversion program for nonviolent offenders.

But Judge Noreika had other ideas, telling the two sides repeatedly that she had no intention of being “a rubber stamp,” and spending three hours sharply questioning them over nearly every detail of the deal.

“I cannot accept the plea agreement today,” said Judge Noreika, who was nominated to the bench by President Donald J. Trump in 2017 with the support of Delaware’s two Democratic senators.

The muddled outcome only underscored how Mr. Biden’s personal and legal troubles have become an entrenched political issue in Washington, where Republicans have long sought to show that his foreign business ventures were aided by, or benefited, President Biden.

Those efforts only intensified as Mr. Trump’s legal troubles have deepened and Republicans in Congress have sought to undercut the president heading into the 2024 election.

Republicans had accused David C. Weiss, the Trump-appointed U.S. attorney in Delaware who was retained by the Biden administration to complete the investigation into Mr. Biden, of cutting a “sweetheart deal” intended to help Democrats.

On Friday, the U.S. attorney general, Merrick Garland, made Mr. Weiss an independent special counsel, giving him autonomy over the decision to prosecute in any district and leaving it up to him “where, when and whether to file charges.” The announcement insulated the Biden administration somewhat from decisions regarding Mr. Biden.

Republicans have long sought to cast the Biden family as corrupt and assailed the proposed deal as far too lenient, citing testimony from two I.R.S. investigators as evidence that the Justice Department had hamstrung the investigation and that President Biden played a role in his son’s business deals with companies and partners in Ukraine and China.

Erica L. Green contributed reporting.

After his father became vice president, Hunter Biden, a Yale-educated lawyer, forged business relationships with foreign interests that brought him millions of dollars, raised questions about whether he was cashing in on his family name, set off alarms among government officials about potential conflicts of interest, and provided Republicans an opening for years of attacks on his father.

And after the death of his brother, Beau, in 2015, Hunter descended into a spiral of addiction and tawdry and self-destructive behavior.

He is sober now and no longer entangled in foreign business deals. He is a visible presence in his father’s life — his oldest daughter was married at the White House in November, and he attended a state dinner last month.

An examination by The New York Times of the investigation and Hunter Biden’s journey to this juncture, based on interviews with his former business partners, family members and close friends, as well as officials and lawyers familiar with the Justice Department investigation, does not provide either side with the narrative they would prefer.

What emerges is the story of a man battling with personal demons against the background of family tragedy and under the glare of public scrutiny. It is also an instructive look at the enduring Washington practice of trading on access and influence.