News International chief executive Tom Mockridge said the company was "keen to play our full part, with others in our industry, in creating a new body that commands the confidence of the public."
"We believe that this can be achieved without statutory regulation — and welcome the prime minister's rejection of that proposal."
Leveson's 4 million pound ($6.4 million) inquiry heard evidence from more than 300 witnesses during months of hearings that provided a dramatic, sometimes comic and often poignant window on the workings of the media. Witnesses ranged from celebrities such as Harry Potter author J.K. Rowling and Hugh Grant — who both complained of intrusive treatment — to the parents of Dowler, who described how learning that their daughter's voicemail had been accessed had given them false hope that she was alive.
Leveson said that the ongoing criminal investigation constrained him from accusing other newspapers of illegal behavior, but concluded there was a subculture of unethical behavior "within some parts of some titles."
While many editors have denied knowing about phone hacking, Leveson said it "was far more than a covert, secret activity, known to nobody save one or two practitioners of the 'dark arts.'"
He said newspapers had been guilty of "recklessness in prioritizing sensational stories almost irrespective of the harm the stories may cause."
"In each case, the impact has been real and, in some cases, devastating," the judge said.
The hacking scandal has rocked Britain's press, political and police establishments, who were revealed to enjoy an often cozy relationship in which drinks, dinners and sometimes money were traded for influence and information.
Several senior police officers resigned over the failure aggressively to pursue an investigation of phone hacking at the News of the World in 2007. But Leveson said that "the inquiry has not unearthed extensive evidence of police corruption."
Leveson said over the past three decades, political parties "have had or developed too close a relationship with the press in a way which has not been in the public interest."
Those relationships reached right up to the prime minister's door. Former Murdoch editors and journalists charged with phone hacking, police bribery or other wrongdoing include Cameron's former spokesman, Andy Coulson, and ex-News of the World editor Rebekah Brooks, a friend of the prime minister.
Leveson acquitted senior politicians of wrongdoing, but recommended that political parties publish statements "setting out, for the public, an explanation of the approach they propose to take as a matter of party policy in conducting relationships with the press."
Cameron, who is tainted by his own ties to prominent figures in the scandal, said he accepted that proposal.
But politicians remained far apart on the broader issue of how, or whether, to regulate the press.
Cameron was holding talks Thursday with leaders of the other main parties in an attempt to thrash out agreement.
He faced a battle. Deputy Prime Minister Nick Clegg, leader of junior government partner the Liberal Democrats, differed from Cameron in backing the call for a new regulator established in law.
"We owe it to the victims of these scandals, who have already waited too long for us to do the right thing," he said.
Analysts say that it was possible for the coalition government's two parties to join forces and push through a version of the recommended legal changes.
But Steven Barnett, a communications professor at the University of Westminster, said that if that does not happen, he would not trust the British press to set up a truly independent regulator.
"One possibility is that in the end (the report) has no effect whatsoever," Barnett said. "The press can make some noises about regulating themselves. But in the end they will want to control themselves in ways that Leveson himself said was unacceptable."