Buckingham Palace said the story was a "serious breach" of privacy
A newspaper has issued an unreserved apology to the Royal Family over a story that the Duke of Edinburgh had been diagnosed with prostate cancer.
The London Evening Standard said the "distressing allegation" was untrue and constituted a breach of privacy.
On Wednesday, it reported the prince, 87, had been diagnosed while being treated for a chest infection in April.
Buckingham Palace welcomed the apology and the Press Complaints Commission said the matter was now settled.
In its apology, the paper said the duke was "not suffering from any such condition".
"We unreservedly apologise both to him and to his family for making this distressing allegation and for breaching his privacy," it added.
After the story appeared, the palace issued a statement saying it was a "serious breach of Prince Philip's privacy" and reported the case to the Press Complaints Commission (PCC).
A Buckingham Palace spokeswoman said: "We welcome the apology from the Evening Standard."
The PCC, an independent watchdog which deals with complaints about the editorial content of newspapers, said the apology meant the matter had been settled without the need for a formal ruling.
A PCC spokesman added: "The process has been quick: the final settlement was negotiated less than 36 hours after the original complaint was made.
"The apology has been prominent and proportionate. And the PCC costs nothing to use."
BBC News royal correspondent Nicholas Witchell said the apology was an "embarrassing climbdown" for the Evening Standard.
He added that Buckingham Palace was anxious to draw "red lines" about what can be reported, and warn the media that they should intrude over those lines "at their peril".
On Wednesday, Buckingham Palace took the unprecedented step of denying the front-page story, entitled "Prince Philip Defies Cancer Scare".
The palace said that although it was its policy not to comment on rumours regarding the health of family members as they had a right to privacy, it had decided to deny the Evening Standard report.
In April, Prince Philip spent three days in King Edward VII's Hospital, west London, where he was treated for a chest infection.