Business Secretary Vince Cable will stay in cabinet despite "declaring war" on Rupert Murdoch, says Downing Street.
But he will be stripped of his powers to rule on Mr Murdoch's bid to take control of BSkyB, which will be handed to Culture Secretary Jeremy Hunt.
Downing Street said David Cameron believed Mr Cable's comments about Mr Murdoch were "totally unacceptable and inappropriate".
Labour leader Ed Miliband said he would have sacked Mr Cable.
BBC Political Editor Nick Robinson said Downing Street's statement was a "humiliating slap in the face" for Mr Cable and although he had kept his job he was still in a "very awkward" position, with questions over his judgement after telling "complete strangers" highly sensitive political information.
Nick Robinson said some Tory MPs would be questioning whether Mr Cable - who is seen as a key figure in holding the coalition together - would have been "out on his ear" if he had been a member of their party.
In a statement, a Downing Street spokesman said: "Following comments made by Vince Cable to the Daily Telegraph, the prime minister has decided that he will play no further part in the decision over News Corporation's proposed takeover of BSkyB.
"In addition, all responsibility for competition and policy issues relating to media, broadcasting, digital and telecoms sectors will be transferred immediately to the Secretary of State for Culture, Media and Sport.
"This includes full responsibility for Ofcom's activities in these areas.
"The prime minister is clear that Mr Cable's comments were totally unacceptable and inappropriate."
Mr Cable said in a statement: "I fully accept the decision of the Prime Minister and Deputy Prime Minister. I deeply regret the comments I made and apologise for the embarrassment that I have caused the government."
News Corporation, which already owns 39% of BSkyB but wants to buy up the remaining 61% for £7.8bn, said Mr Cable's unguarded remarks raised "serious questions about fairness and due process".
As business secretary, Mr Cable was to have the final say over whether the takeover should be allowed to go ahead amid concerns about press freedom and consumer choice.
But Labour claimed his impartiality was fatally undermined by comments made to undercover Daily Telegraph reporters posing as his constituents.
Labour leader Ed Miliband replied yes when asked if he would have sacked Mr Cable had he been in Mr Cameron's position.
"Having apparently breached the ministerial code and having said what he said, he shouldn't be remaining in office and I fear that David Cameron has made this decision not because it's good for the country, but because he is worried about the impact on his coalition of Vince Cable going.
"That's not the way decisions about who is in and out of government should be made."
The comments were not included in a transcript of Mr Cable's conversation with the undercover reporters published earlier on Tuesday by the newspaper.
They were leaked to BBC Business Editor Robert Peston by a source who was upset at the newspaper's decision not to publish them.
According to the leaked transcript, Mr Cable said: "I am picking my fights, some of which you may have seen, some of which you may haven't seen.
"And I don't know if you have been following what has been happening with the Murdoch press, where I have declared war on Mr Murdoch and I think we are going to win."
He also speaks of the importance of not politicizing the BSkyB decision "because it is a legal decision".
But adds: "I have blocked it using the powers that I have got and they are legal powers that I have got. I can't politicise it but from the people that know what is happening this is a big, big thing.
"His whole empire is now under attack... So there are things like that we do in government, that we can't do... all we can do in opposition is protest."
Mr Cable had earlier revealed deep misgivings about a number of coalition policies in his secretly taped remarks - including bank bonuses, immigration, and the speed at which the coalition was trying to push through changes in the health service, local government and other areas.
He also claimed he could "bring the government down" if he resigned and this was the "nuclear option" he was keeping in reserve as a last resort.
The newspaper said it planned to publish further extracts from the tapes in the coming days.
But writing in his blog, Robert Peston suggested the decision not to publish the Murdoch remarks may have been commercially driven.
He said: "The Telegraph has been a leading opponent of News Corporation's attempt to acquire the whole of BSkyB. In October, the Telegraph's chief executive, Murdoch MacLennan, signed a letter - along with senior executives of the BBC, Channel 4, the Daily Mail and Trinity Mirror - asking Mr Cable to consider blocking the takeover.
"The disclosure of Mr Cable's private views on Mr Murdoch and the proposed takeover of BSkyB makes it extremely difficult for him to fulfil his role as the ultimate arbiter of whether the deal should proceed under the 2002 Enterprise Act.
"News Corporation is bound to challenge his impartiality."
It comes as News Corporation's bid for full control of BSkyB was cleared by the European Commission.
Ofcom must decide by 31 December whether to refer the bid to the Competition Commission - but the final decision on blocking it will now rest with Culture Secretary Jeremy Hunt.