London (CNN) -- Media magnate Rupert Murdoch's flagship British Sunday tabloid newspaper officially apologized Sunday for hacking into voice mails, in a scandal which has affected celebrities, politicians and royal household staff.
The weekly newspaper offered compensation and "apologized unreservedly" for the "unacceptable" hacking.
But at least one of the victims is rejecting the deal, her lawyer told CNN Sunday.
Nicola Phillips, who used to work for celebrity publicist Max Clifford, has refused a payout offer, lawyer Mark Lewis said.
"She ... needs a declaration of the truth," Lewis told CNN. "If I said I will give you 50,000 pounds ($82,000), but you have no way of knowing how many times your phone was hacked, you have no way of knowing how much damage was done."
He declined to say how much she had been offered, citing the terms of a letter Phillips received on Friday from News International, the News of the World's parent company.
She is one of several Lewis clients suing over the scandal. Lewis said none of his other clients had received offers of compensation. He represents "Who Wants to Be a Millionaire" host Chris Tarrant, among others.
News International, which also owns the Sun and the Times of London, Friday apologized for the widespread phone hacking scandal.
Murdoch's empire also encompasses the Wall Street Journal, the New York Post, Fox News and Harper Collins publishers.
Actress Sienna Miller on April 5 won a court case to access phone records to see if her phone had been hacked.
That same day, London's Metropolitan Police arrested two men "on suspicion of unlawfully intercepting mobile phone voice mail messages" on April 5, they said. British media identified them as Ian Edmonson, a former assistant editor of News of the World, and Neville Thurbeck, its chief reporter.
By law, British police do not name people who have been arrested.
Former British Prime Minister Gordon Brown wrote to police last summer to ask if his voice mail had been hacked into, a source close to the situation told CNN in January. Police declined to say whether Brown had written to them or what action they took. A spokeswoman for Brown would not comment on the record.
News International issued a similar apology Friday: "Following an extensive internal investigation and disclosures through civil legal cases, News International has decided to approach some civil litigants with an unreserved apology and an admission of liability in cases meeting specific criteria."
The company added that it has asked its lawyers to set up a compensation fund to deal with "justifiable" claims -- but would contest cases that it believed were without merit.
"That said, past behaviour at the News of the World in relation to voice mail 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 then were not sufficiently robust.
"We continue to co-operate fully with the Metropolitan Police. It was our discovery and voluntary disclosure of this evidence in January that led to the re-opening of the police investigation.
"With that investigation on going, we cannot comment further until its completion."
In January, Prime Minister David Cameron's spokesman stepped down amid speculation he had known about the hacking -- something he has always denied.
Andy Coulson was the editor of the News of the World in 2007 when royal correspondent Clive Goodman and private investigator Glenn Mulcaire were sentenced to prison for hacking into voice mails of members of the royal family's staff.
Goodman also admitted hacking into the messages of model Elle MacPherson.
Coulson denied knowing about the hacking but resigned from the paper because Goodman ultimately worked for him.
He was later hired to be Cameron's spokesman, a position he quit earlier this year after an extensive New York Times investigation reignited the scandal.
Cameron said Coulson was stepping down because of the continuing pressures on him and his family. The former newspaper editor also believed that the focus surrounding the issue was "impeding his ability to do his job and was starting to prove a distraction for the Government," Cameron added.
The New York Times alleged in its story last year that -- far from Goodman and Mulcaire being lone culprits -- phone hacking was common practice at the newspaper.
One of the few sources who went on the record in the Times article, former News of the World journalist Sean Hoare, said Coulson, then his boss at the tabloid, "actively encouraged me" to hack into the voice mails of public figures to get stories for the News of the World.
But Coulson's allies have cast doubt on Hoare's credibility since the Times article came out September 1, pointing out that Hoare was fired from the paper over allegations of drug and alcohol abuse.
A British parliamentary committee has also investigated the tabloid. News Corporation executives insisted in parliamentary testimony that illegal hacking into voice mail messages was not a widespread practice at News of the World.
CNN's Nick Hunt contributed to this report.