Deputy Attorney General Rod Rosenstein appointed a former FBI director as special counsel to oversee the investigation into Russian efforts to influence the presidential election, agreeing to Democrats’ demands to elevate the probe and put it outside President Trump’s political chain of command.

Mr. Rosenstein tapped Robert Mueller, whose 12-year tenure at the head of the FBI earned him praise from both Democrats and Republicans who said he is above reproach.

The deputy attorney general, who inherited responsibility for the investigation after Attorney General Jeff Sessions recused himself, said it was “in the public interest” to name a special counsel — though he said that shouldn’t be taken as a suggestion of any wrongdoing.

“My decision is not a finding that crimes have been committed or that any prosecution is warranted. I have made no such determination,” Mr. Rosenstein said in a statement. “What I have determined is that based upon the unique circumstances, the public interest requires me to place this investigation under the authority of a person who exercises a degree of independence from the normal chain of command.”

Mr. Trump predicted he would be exonerated.

“As I have stated many times, a thorough investigation will confirm what we already know: There was no collusion between my campaign and any foreign entity,” he said. “I look forward to this matter concluding quickly. In the meantime, I will never stop fighting for the people and the issues that matter most to the future of our country.”

Pressure had grown to appoint a special counsel after Mr. Trump last week fired FBI Director James B. Comey, who succeeded Mr. Mueller at the bureau in 2013.

Mr. Mueller will step down from his private law firm to avoid any conflicts of interest during the process. He will also have the authority to investigate any links or coordination between the Russian government and associates of the Trump presidential campaign as well as any other matter that arises directly from the investigation.

“I accept this responsibility and will discharge it to the best of my ability,” he said in a statement.

Democrats called the choice of Mr. Mueller inspired.

“Former Director Mueller is exactly the right kind of individual for this job,” said Senate Minority Leader Charles E. Schumer, New York Democrat. “I now have significantly greater confidence that the investigation will follow the facts wherever they lead.”

Some Democrats said one of Mr. Mueller’s first steps should be to expand the Russia probe to include looking into the firing of Mr. Comey, particularly after reports this week that the president had asked the FBI chief to back off the investigation into former National Security Adviser Michael Flynn’s ties to foreign governments.

Even as the criminal investigation proceeds, Democrats said Congress must pursue its own multiple investigations.

Leaders from the Senate Select Committee on Intelligence invited Mr. Comey and acting FBI Director Andrew McCabe to testify again, both publicly and behind closed doors.

They also asked the country’s leading law enforcement agency to hand over all correspondence from Mr. Comey, the White House and the Justice Department regarding the FBI’s Russia investigation.

Senate Judiciary Committee members requested similar records.

“We need all those memos,” said Chairman Chuck Grassley, Iowa Republican. “All communications between everybody — Comey, and other people in the hierarchy.”

On Tuesday night, Rep. Jason Chaffetz, Utah Republican and chairman of the House Oversight and Government Reform Committee, asked the FBI to provide copies of all agency memos detailing Comey-Trump correspondence. He said he has his subpoena pen ready if his request is ignored.

While some cracks have begun to appear among Republicans, party leaders said Wednesday that they were waiting for more information.

“We can’t deal with speculation and innuendo, and there’s clearly a lot of politics being played,” House Speaker Paul D. Ryan, Wisconsin Republican, said after a morning meeting with his colleagues. “We need the facts.”

He was reacting to a flurry of news reports that included accusations that Mr. Trump shared highly sensitive information with the Russian foreign minister in an Oval Office meeting this month and that he asked Mr. Comey to stop investigating Mr. Flynn after his resignation as national security adviser.

Mr. Mueller became FBI director under President George W. Bush on Sept. 4, 2001, a week before the terrorist attacks, and helped shift the bureau to an anti-terrorism mission.

His term was set to expire in 2011, but President Obama asked him to stay on while the administration sought a credible replacement — ending up with Mr. Comey. His tenure as FBI director was the longest since J. Edgar Hoover.

Mr. Mueller, now 72, has been working as a lawyer for the WilmerHale firm. He was also tapped last year by Booz Allen Hamilton to conduct an external review of its security, personnel, and management processes and practices after a former National Security Agency contractor employed was accused of taking more than a half-billion pages of top-secret documents.

As U.S. attorney for the District of Maryland, Mr. Rosenstein’s office was overseeing the charges against former National Security Agency contractor Harold Martin.

Democrats demanded Wednesday that the Justice Department agree to give Mr. Mueller the funds needed to hire a staff and pursue the probe aggressively. Under the law, he is allowed to request a budget from the department and is supposed to be granted “all appropriate resources.”

Mr. Rosenstein said he will make sure Mr. Mueller has those resources.

Special prosecutors have the authority to exercise all investigative and prosecutorial functions of a typical prosecutor — including the power to subpoena witnesses and bring criminal charges. They are allowed to choose their own employees and request a budget from the Justice Department but are not subject to day-to-day supervision by the department.

Two other Wilmer Hale attorneys — James Quarles and Aaron Zebley — have also resigned from the firm to join Mr. Mueller in the investigation.

During his own confirmation hearing, Mr. Rosenstein faced questions about whether he would appoint special counsel to handle the Russia investigation.

He demurred at the time, saying he had no knowledge of the facts of the case and was not aware of anything that would disqualify him from overseeing it. But he said he would be open to the consideration “whenever I determine that it’s appropriate based on the policies and procedures of the Justice Department.”

He appointed Mr. Mueller the day before Mr. Rosenstein is set to brief the Senate on Mr. Comey’s firing and will likely take much of the sting out of what otherwise was setting up to be a painful encounter.

• Tom Howell Jr. contributed to this report.

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