"The Duke of York has been appalled by the recent reports of Jeffrey Epstein's alleged crimes," Buckingham Palace said in a statement on Monday.
"His Royal Highness deplores the exploitation of any human being and the suggestion he would condone, participate in or encourage any such behaviour is abhorrent," the statement said.
The grainy video published by the Mail on Sunday claims to show the prince waving goodbye to a woman as she leaves Epstein's New York mansion in 2010.
Epstein had pleaded guilty in 2008 to procuring for prostitution a girl under the age of 18 and served 13 months in prison before being released on probation.
The video was the latest in a string of revelations about Prince Andrew's friendship with Epstein in British tabloids.
The Mail on Monday said the prince had hosted Epstein at Balmoral, Queen Elizabeth II's home in Scotland, in 1999 at a time when the monarch may have been in residence.
He also reportedly hosted Epstein at Sandringham, another royal residence in eastern England, a year later.
Epstein was found dead on August 10 in a federal jail in Manhattan as he awaited trial charged with one count of sex trafficking of minors and one count of conspiracy to commit sex trafficking of minors.
A coroner ruled that he committed suicide by hanging.
Over the years, the financier hobnobbed with politicians, socialites and celebrities, including Donald Trump before he was president, and Bill Clinton.
Virginia Giuffre -- previously Virginia Roberts -- one of Epstein's alleged victims, has testified that she was forced to have sex with Prince Andrew in London when she was 17. She said she had sex with him again in New York and on Epstein's private island in the Caribbean.
But the allegations were struck from the record by a US judge in 2015, who said the "lurid details" were not needed to decide a civil case concerning Epstein.
Buckingham Palace has repeatedly and strenuously denied the claims, calling them "false and without foundation".
Prince Andrew briefly referred to the allegations at the Davos annual meeting in 2015, saying: "I wish to reiterate and reaffirm the statements already made on my behalf by Buckingham Palace. My focus is my work."
The prince was Britain's special representative for international trade and investment for 10 years until 2011, when he gave up the post following intense scrutiny over his relationships with Epstein and other controversial figures.
The prince plays a minor role in royal affairs.
In 2014, he set up an initiative called Pitch@Palace to bring together tech entrepreneurs.
He is currently reported to be on holiday in Spain.