Cable will remain in post as business secretary despite the explosive outburst, which referred to his decision to intervene in the bid by Murdoch's News Corp to take full control of Sky on grounds of public interest.
Last night, Cable was called to an emergency meeting at Downing Street, wherein he was told that he will play no further part in the takeover decision, which will transfer to Hunt.
Downing Street said that Prime Minister David Cameron viewed Cable's comments - secretly recorded by reporters for the Daily Telegraph - as "totally unacceptable and inappropriate."
In a statement, Cable said: "I fully accept the decision of the prime minister and deputy prime minister (Nick Clegg). I deeply regret the comments I made and apologise for the embarrassment that I have caused the government."
News Corporation, which wants to buy the 61% of Sky that it does not already own for £7.8 billion, said Cable's outburst raised "serious questions about fairness and due process".
Labour leader Ed Miliband called for Cable to be sacked from the Cabinet as he had breached the ministerial code on objectivity in decision-making.
"David Cameron has made the wrong judgment and he has kept Vince Cable on, not because of the national interest but because his Conservative-led government needs the prop which Vince Cable provides," he said.
"Vince Cable should have gone. Having apparently breached the ministerial code and having said what he said, he shouldn't be remaining in office."
Ofcom must submit its report on the Sky takeover by December 31, when Hunt will have to decide whether to refer the bid to the Competition Commission.