Deputy Attorney General Rod Rosenstein said it was in the public interest to bring in an outsider.
The appointment has been widely praised by politicians from both sides.
Calls for a special prosecutor had mounted since President Donald Trump fired the most recent FBI director, James Comey, last week.
The FBI and Congress are looking into potential links between Mr Trump’s campaign team and Russia.
Just over an hour after the news of Mr Mueller’s appointment emerged, President Trump predicted the investigation would confirm there had been no collusion.
“A thorough investigation will confirm what we already know – there was no collusion between my campaign and any foreign entity,” he said.
The top Democrat in the Senate, Chuck Schumer, said Mr Mueller was “exactly the right kind of individual for this job”.
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And the House Oversight Committee Chairman Jason Chaffetz said he had “impeccable credentials”.
In his statement announcing the move, Mr Rosenstein said: “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.”