Mueller will have sweeping powers, including the right to bring federal charges.
The surprise announcement to hand the probe over to Mueller, a lawman who garners deep bipartisan respect, was a striking shift for Trump's Justice Department, which had resisted increasingly loud calls from Democrats for an outside prosecutor. "Mr. Rosenstein has done the right thing", Senate Minority Leader Chuck Schumer of NY said Thursday on the Senate floor. "I now have significantly greater confidence that the investigation will follow the facts wherever they lead".
In a tweet on May 12, Trump said: "Russia must be laughing up their sleeves watching as the United States tears itself apart over a Democrat EXCUSE for losing the election".
Democrats and Trump's fellow Republicans widely praised the appointment of a special counsel to take over the Russian Federation probe.
So far, no evidence of collusion has been presented, but the question has consumed the first four months of Trump's presidency and spurred multiple crises that have left the White House shell-shocked.
Democrat El Green has demanded impeachment of Trump.
Mueller's broad mandate gives him not only oversight of the Russian Federation probe, but also over "any matters that arose or may arise directly from the investigation".
Trump fired Comey last week and wrote in a letter to the former director that he had told Trump on three separate occasions he was not under investigation.
Mueller, a former federal prosecutor at the Justice Department, was confirmed as FBI director days before the September 11, 2001, attacks that would ultimately shape his tenure.
Comey succeeded him, appointed by Obama.
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Rosenstein had faced criticism as the author of a memo that preceded Comey's firing.
Democratic Senatr Al Frankens said that appointing Mueller is an important step that will hopefully help restore the public trust and result in a thorough and comprehensive investigation.
"We owe it to the American people to find out what happened, and I'm encouraged to see that the justice department has appointed a special counsel to lead the investigation into Russia's interference in the 2016 election", Indian-American Congressman Ami Bera said.
As for Comey, whom Trump fired last week, the FBI director wrote in a memo after a February meeting at the White House that the new president had asked him to shut down the FBI's investigation of Flynn and his Russian contacts, said a person who had read the memo.
Cutting across party lines, top American lawmakers welcomed the Mueller's for the investigation.
The White House has played down the importance and secrecy of the information Trump gave to the Russians, which came from Israel under an intelligence-sharing agreement.
"And at least one senior staffer has begun privately talking to friends about what a post-White House job would look like, according to two people close the staffer".
"The appointment of former FBI director and respected lawyer Robert Mueller as special counsel for the Russian Federation investigation is a positive development and will provide some certainty for the American people that the investigation will proceed fairly and free of political influence", Senator Richard Burr said and Mark Warner, chairman and ranking member of Senate Select Committee on Intelligence, said. The others were former Connecticut Sen. Joe Lieberman, former Oklahoma Gov.
Trump is leaving Friday for his first foreign trip, to the Mideast and beyond, and aides had hoped the disarray at home would have been calmed if not resolved by take-off time.