The paper, which is being sued by a number of celebrities including Sienna Miller, revealed that parent company News International will establish a compensation scheme to "deal with justifiable claims fairly and efficiently".
"This will begin the process of bringing these cases to a fair resolution with damages appropriate to the extent of the intrusion," the statement said.
"We will, however, continue to contest cases that we believe are without merit or where we are not responsible. That said, past behaviour at the News of the World in relation to voicemail interception is a matter of genuine regret. It is now apparent that our previous inquiries failed to uncover important evidence and we acknowledge our actions were not sufficiently robust.
"News International's commitment to our readers and our pride in our award-winning journalism remains undiminished."
The statement added that the paper is fully cooperating with a review being conducted by the Press Complaints Commission.
News International announced on Friday that it will admit liability in a number of alleged phone hacking cases brought against the newspaper.