After cross-party talks this evening, a senior Labour source has told the Press Association that David Cameron had agreed to ask the Department for Culture, Media and Sport to draft a bill to implement Leveson's recommendations.
The source said that the talks lasted less than 30 minutes and were "frank and to the point."
A man who won libel damages after his arrest over the murder of a woman has written to MPs asking them to back Leveson's recommendations.
About 2,000 editors will sign up to a new independent press watchdog, the chairman of the Press Complaints Commission has said.
The Prime Minister has taken a political gamble by holding differing views to the victims, the Leveson inquiry and a majority of MPs.