Prince Harry and six other high-profile figures in Britain were ordered on Friday to make an initial payment of £9.54 million, about $13 million, to the publisher of The Daily Mail, after they lost a privacy case against the company.

In July the group of claimants, which included the singer Elton John and the actress Liz Hurley, failed in its efforts to sue the publisher, Associated Newspapers, over allegations of phone hacking and intruding into its members’ private lives between 1993 and 2011.

On Friday a judge in the High Court in London ordered them to make an initial payment “on account” by Aug. 28 for the legal costs incurred by the newspaper company in the case.

In a written statement that summarized the ruling, the judge, Sir Matthew Nicklin, said that “the reasonable sum on account is £9,544,355.”

The judge, whose formal title in the High Court is Mr. Justice Nicklin, declined to impose a cap on the total payment that might be ordered from the claimants, meaning that they might eventually have to pay up to £34.5 million, the costs that Associated Newspapers said it had incurred up to July this year.

However, he said in his full ruling that Associated’s costs “appear exceptionally high” and “largely unexplained,” so a court assessment would take place to determine the appropriate final sum.

The judge’s ruling comes just days after Prince Harry and his wife, Meghan, made public their plans to return to Britain, with their children Prince Archie, 7, and Princess Lilibet, 5. The court case against Associated Newspapers was Harry’s last in a series he has undertaken against Britain’s tabloid newspapers, with whom he has had a fractious relationship.

The seven claimants ordered to pay the initial multimillion-pound fee are Harry; Elton John and his husband, David Furnish; Simon Hughes, a former lawmaker; the actresses Sadie Frost and Liz Hurley; and Doreen Lawrence, a member of the House of Lords and social justice campaigner whose son was the victim in a notorious 1993 racist murder case.

The seven claimants had accused private investigators of listening in on phone calls, intercepting voice mail messages and tricking people into handing over information to obtain phone numbers, medical records and bills.

During the trial, those who gave evidence in court included Harry, who flew in from the United States to attend the hearing in person. Current and former employees of Associated also appeared, denying any wrongdoing.

In a ruling in July, the judge ruled that Harry and the other high-profile claimants had failed to prove “that the information complained of had been obtained unlawfully.”

In his statement on Friday, the judge rejected a request from the group of claimants to set a cap on costs at between £18 million and £20 million, arguing that doing so would “risk being no more than a judicial estimate of what appears, at this stage, to be a reasonable maximum.”

He added: “That would be vulnerable to the charge that it was arbitrary. It would also risk unfairness to Associated, because it would deprive Associated of the opportunity to demonstrate, on evidence and in the proper forum, that costs above the cap were reasonably incurred.”

However, he made clear his skepticism about the scale of the costs that the newspaper group had claimed. “I regard a claim for costs in excess of £34 million as, on its face, excessive and as giving rise to real concerns as to whether all of the costs now claimed by Associated were reasonably incurred and are reasonable in amount,” he wrote.

According to The Law Society Gazette, a British legal publication, the group of celebrities that brought the claim have legal insurance covering them for £16.2 million in the event of them losing the case.