The judge said that his proposals would protect the rights of victims and people bringing complaints, but also guarantee a free press.
[Leveson with the report inside the QEII Conference Centre in Central London]
The Leveson inquiry was launched following public revulsion at claims that the News of the World had hacked the voicemail of murdered schoolgirl Milly Dowler in 2002.
It expanded to a wider examination of the press, politicians and the police, hearing testimony from former prime ministers, celebrities and media barons, such as Rupert Murdoch.
Lord Justice Leveson said that the press is one of the "true safeguards" of our democracy, but such "power and influence" carries responsibilities that have not been upheld.
He was very critical of the relationship between UK politicians and press over last 20 years, and said that "without doubt" the press' own code of conduct has been damaged and ignored over the years, resulting in activity that has "wreaked havoc with the lives of innocent people whose rights and liberties have been disdained".
Lord Leveson said that the current self-regulation body, the Press Complaints Commission, had "failed", and there was need for "change".
[British actor Hugh Grant, who testified to the Leveson Inquiry, arrives]
He called for a "genuinely independent and effective" system of self regulation of standards, but said that a proposal put forward by Lord Hunt and Lord Black, who currently run the PCC, for a new model of self-regulation, did not go far enough.
"Any model with editors on the main board is simply not independent... it is still the industry marking its own homework," he said.
Lord Leveson said that the press needs to set up an independent body that is free of industry figures and politicians. The chair and other members of the new regulator must be independent and appointed in a fair way, and the body must not include any serving editor or politician.
His report recommends a system of compulsory arbitration that would allow victims of alleged press intrusion to seek apologies and compensation without having to go through the courts.
Lord Justice Leveson said that this new system of self-regulation cannot be established without legislation, but added: "This is not, and cannot, and should not reasonably be characterised as a statutory process of regulating the press."
[Bob and Sally Dowler, the parents of murdered schoolgirl Milly Dowler, provided testimony to the Leveson inquiry]
In the executive summary of his report, he elaborated: "It is worth being clear what this legislation would not do. The legislation would not establish a body to regulate the press: it would be up to the press to come forward with their own body that meets the criteria laid down.
"The legislation would not give any rights to Parliament, to the Government, or to any regulatory (or other) body to prevent newspapers from publishing any material whatsoever.
"Nor would it give any rights to these entities to require newspapers to publish any material except insofar as it would require the recognised self-regulatory body to have the power to direct the placement and prominence of corrections and apologies in respect of information found, by that body, to require them."
Lord Justice Leveson said that the legislation - which could be included in next year's Communications Bill - would simply "enshrine, for the first time, a legal duty on the government to protect the freedom of the press".
It would also provide "an independent process to recognise the new self-regulatory body and reassure the public that the basic requirements of independence and effectiveness were met and continue to be met". He recommended that this process be overseen by media regulator Ofcom.
[Kate McCann arrives for the announcement]
Lord Justice Leveson said that he wanted the industry to sign up to a legally-binding arbitration process that would force newspapers and magazines to deal effectively with complaints.
The new independent body would have the power to levy "sanctions" on publishers, but Ofcom could be used as a 'backstop regulator' for any publishers that refuse to sign up.
Ofcom already regulates TV and telecoms companies and organisations.
Prime Minister David Cameron, who set up the Leveson Inquiry, will make a statement to the Parliament on the report at 3pm.
But his deputy, Nick Clegg, is to make his own statement in the Commons, after reportedly failing to agree with the prime minister's stance on future press regulation.