Defence Secretary Michael Fallon is to pay undisclosed compensation to imam Suliman Gani for falsely claiming he supported Islamic State.
The Cabinet Minister announced the move on his website as he apologised to the Muslim leader and agreed to pay his legal costs.
Prime Minister David Cameron has also apologised for saying in the House of Commons that Mr Gani supported IS during an attack on then London mayoral candidate Sadiq Khan for sharing a platform with the imam.
Mr Fallon's letter to Mr Gani states: "On 7 May 2016 I was interviewed on BBC Radio 4's Today programme. During that interview I described you as a supporter of Daesh/Islamic State - something that is entirely untrue.
"When I did this I knew that it had been said on a BBC television programme some weeks earlier but I was unaware that the BBC had corrected that statement and had apologised to you. Had I known of that correction and apology I would not have repeated the statement.
"I was made aware of the BBC's correction and apology a few hours after the broadcast and immediately issued a statement in an effort to put the record straight. I issued a further statement to the same effect on 11 May 2016, which included my apology to you for repeating the untrue statement.
"I accept that you are entirely opposed to Daesh/Islamic State, that you regard it as incompatible with your religious and moral beliefs, and that you have spoken out publicly against it.
"I repeat my apology for the error that I made and for the distress that it caused to you and your family.
"In recognition of that distress I have agreed to make a payment of compensation and to meet your reasonable legal costs."