Virginia Giuffre is one of 16 women who say they were abused by convicted paedophile Epstein, and she has previously claimed in court papers that she was forced to have sex with Andrew as a 17-year-old.
Her allegations, which the prince strongly denies, were struck from US civil court records in 2015 after a judge said they were “immaterial”.
In her first broadcast interview along with five other alleged victims, Mrs Giuffre said she was forced into sexual encounters with the prince in London, Epstein’s New York mansion, and a third incident in the US Virgin Islands.
Buckingham Palace has previously contested reports about the prince’s relationship with Epstein, including a private jet flight they are said to have shared with Mrs Giuffre, then known as Virginia Roberts.
Mrs Giuffre said she was introduced to Epstein, and later Prince Andrew, by British socialite Ghislaine Maxwell.
She said: “The first time in London I was so young. Ghislaine woke me up in the morning and said: ‘You’re going to meet a prince today’.
“I did not know at that point I was going to be trafficked to that prince, and then that night Prince Andrew came to her house in London.”
Mrs Giuffre said she spent a night in Mayfair’s exclusive Club Tramp with Prince Andrew, Epstein and Ms Maxwell.
She continued: “We leave Club Tramp and I hop in the car with Ghislaine and Jeffrey, and Ghislaine said ‘he’s coming back to the house and I want you to do for him what you do for Epstein’.
“I couldn’t believe it.”
Describing Prince Andrew after the alleged incident, she said: “He wasn’t rude or anything about it, he said thank you and some kind of soft sentiments like that, and left.
“I couldn’t believe that even royalty were involved.
“He denies that it ever happened, and he is going to keep denying that it ever happened, but he knows the truth and I know the truth.”
Disgraced financier Epstein was found dead on 10 August 10 in his New York prison cell while awaiting trial on sex abuse charges.
Buckingham Palace said it had nothing to add to its previous statement.
“It is emphatically denied that the Duke of York had any form of sexual contact or relationship with Virginia Roberts,” the palace said. “Any claim to the contrary is false and without foundation.”