News of the acceptance that it had "breached a legal obligation" came as two judges in London heard that a number of claimants - including former Deputy Prime Minister Lord Prescott - had settled judicial review proceedings brought against the Met over "failures to warn victims".
Lord Justice Gross and Mr Justice Irwin were told that the two sides had reached agreement by Hugh Tomlinson QC, representing Lord Prescott, ex Met Police Deputy Assistant Commissioner Brian Paddick, actor Jude Law's personal assistant Ben Jackson, MP Chris Bryant and an anonymous individual known as HJK.
Mr Tomlinson said the claimants and the Met had agreed a "declaration" - in which the Met admits it breached its duties under Article 8 of the European Convention on Human Rights.
Lord Prescott was in court for the proceedings.
Law firm Bindmans, for the claimants, said in a statement that the declaration "constitutes an admission by the police that their failure to warn victims that their privacy was or may have been unlawfully invaded was a breach of Article 8 of the European Convention on Human Rights".
That article provides that "everyone has the right to respect for his private and family life, his home and his correspondence".
Lord Prescott said in a statement: "It's taken me 19 months to finally get justice.
"Time and time again I was told by the Metropolitan Police that I had not been targeted by Rupert Murdoch's News of the World.
"But I refused to accept this was the case. Thanks to this judicial review, the Metropolitan Police has finally apologised for its failure to inform victims of the criminal acts committed by the News of the World against myself and hundreds of other victims of phone hacking."
The Metropolitan Police Service said in a statement: "The MPS is pleased to have reached an agreement in this case and accepts more should have been done by police in relation to those identified as victims and potential victims of phone hacking several years ago.
"It is a matter of public record that the unprecedented increase in anti-terrorist investigations resulted in the parameters of the original inquiry being tightly drawn and officers considered the prosecution and conviction of Clive Goodman and Glen Mulcaire as a successful outcome of their investigation.
"There are now more than 130 officers involved in the current phone-hacking inquiry (Weeting) and the two operations being run in conjunction with it, and this in part reflects the lessons that have been learned about how police should deal with the victims of such crimes.
"Today's settlement does not entail damages being paid by the MPS and, as the court has made clear, sets no precedent for the future.
"How the MPS treats victims goes to the very heart of what we do. It was important that this case did not result in such a wide duty being placed on police officers that it could direct them away from their core purpose of preventing and detecting crime.
"All the claimants are receiving personal apologies from the MPS."
MP Chris Bryant said: "I am delighted that the Metropolitan Police are finally admitting that they should have notified not just me but all the thousands of victims of the News of the World's criminality."
He said: "It's a sadness that it has taken all this time to get the Met to admit that they should have notified all the victims - and that we had to go to court to secure that admission."
The claimants' solicitor, Tamsin Allen, of Bindmans, said that at the time of the first investigation into phone hacking, "instead of warning the hundreds or thousands of victims of voicemail interceptions, the police made misleading statements which gave comfort to News International and permitted the cover-up to continue".
She added: "If the police had complied with their obligations under the Human Rights Act in the first place, the history of the phone-hacking scandal would have been very different."
Mr Bryant said: "It is still a complete mystery to me why the police failed properly to investigate the News of the World in 2006, why they failed to examine the material they had garnered from Mr Mulcaire, why they continued to tell parliament that they had contacted all the victims when they hadn't, why they refused to show me the material that related to me and why they refused to reopen the investigation even when
there was clear evidence that the original investigation had only scratched the surface of the criminality at News International.
"I hope Operations Weeting, Tuleta and Elvedon lead to charges and prosecutions soon. It is just a shame that we had to go to court to force the Met to do their duty."
Bindmans said the judicial review was launched in September 2010 and the claim was then "vigorously defended" by the Met.
In a statement it said that "following the new police investigation into phone hacking,
and revelations about the evidence in the hands of the police at the time of the first investigation, the police have finally accepted that they breached a legal obligation to warn the phone-hacking victims".
In separate proceedings brought at the High Court by a number of well-known people against News International subsidiary News Group Newspapers, publisher of the now defunct News of the World, Lord Prescott and Mr Bryant accepted £40,000 and £30,000 respectively.
Ben Jackson has accepted £40,000 and HJK £60,000.
PA
