David Cameron has said he is convinced a new press regulator will "work and endure" as newspapers considered whether to sign up to the royal charter.
Media organisations are deciding how to respond to proposals agreed by the main political parties, amid deep divisions over the implications for Britain's centuries-old tradition of press freedom.
Some of the biggest national newspaper publishers warned last night that there are a number of "deeply contentious issues" surrounding the scheme unveiled in Parliament, which have still to be resolved.
But the Prime Minister said: "I'm confident that we've set up a system that is practical, that is workable, it protects the freedom of the press, but it's a good, strong self-regulatory system for victims, and I'm convinced it will work and it will endure."
Mr Cameron was visiting a nursery in Wandsworth, south London - along with Deputy Prime Minister Nick Clegg - to meet parents to discuss an announcement on childcare funding.
The Liberal Democrat leader said: "I think the whole design of Leveson was based on self-regulation, so not something imposed by anybody else, but independent self-regulation, but with a number of incentives which make it worth newspapers actually joining in the system.
"And that's all to do with this detailed business of exemplary costs and damages which can be imposed upon newspapers that don't participate in the system. I hope that when they examine the fine print, they will see that the incentives are strong and that it's worthwhile, not least in order to restore public trust in the conduct of newspapers, for them to join in with the system, and I very much hope they will."
There was outright condemnation from the Newspaper Society, representing the UK's 1,100 local newspapers, which warned that the plans - including fines of up to £1 million - would place a "crippling burden" on local press.
"A free press cannot be free if it is dependent on and accountable to a regulatory body recognised by the state," said Newspaper Society president Adrian Jeakings.
Meanwhile, the 57-nation Organisation for Security and Co-operation in Europe (OSCE) urged Britain not to abandon a tradition of press self-regulation regarded around the world as best practice.