In an announcement today, the owner of the tabloid said it will make an "unreserved apology" to some of the claimants taking civil action against the title, in cases meeting "specific criteria".
It said a compensation fund will also be established by its lawyers with a view to dealing with "justifiable claims fairly and efficiently".
"Past behaviour at the News of the World in relation to voicemail interception is a matter of genuine regret," News International said today.
"It is now apparent that our previous inquiries failed to uncover important evidence and we acknowledge our actions then were not sufficiently robust."
According to the BBC's business editor Robert Peston, News International believes most compensation claims will be settled for less than £100,000 each, with a total pay out of less than £20 million.
Today's statement follows an announcement by Scotland Yard earlier this week that police investigating phone hacking at the newspaper had arrested and bailed two men on suspicion of unlawfully intercepting mobile phone voicemail messages.
The arrests followed the relaunching of the Metropolitan Police's inquiry into phone hacking by the recently formed Specialist Crime Directorate 'Operation Weeting' team in January after receiving "significant new information".
Scotland Yard refused to name the arrested pair, but they were widely reported to be News of the World chief reporter Neville Thurlbeck and former head of news Ian Edmondson. Edmondson was suspended by the Sunday paper in January over a "serious allegation" of phone-hacking.
News International, which also owns the Times, the Sunday Times and the Sun, issued a statement at the time saying that it does not tolerate wrong-doing and "is committed to acting on evidence".
It added today that it would continue to co-operate with the Metropolitan Police, but would also "continue to contest cases that we believe are without merit or where we are not responsible".
In 2007, private investigator Glenn Mulcaire and royal correspondent for the News of the World Clive Goodman were convicted of conspiracy to intercept telephone calls.
See the full News International statement at this link.