Answering questions about his links to convicted sex offender Jeffrey Epstein in a BBC interview, Prince Andrew said the alleged incidents "never happened".
Virginia Giuffre, one of Epstein's accusers, claimed she was forced to have sex with the prince three times.
The prince said he was at home with his children on one of the occasions.
Prince Andrew, who is the Queen's third child, has been facing questions for several months over his ties to Epstein, a 66-year-old American financier who took his own life while awaiting trial on sex-trafficking charges.
Virginia Giuffre - then called Virginia Roberts - has said she was forced to have sex with Prince Andrew between 2001 - when she was 17 - and 2002, in London, New York and Epstein's private island in the US Virgin Islands.
Speaking to BBC Newsnight's Emily Maitlis, the prince said: "It didn't happen. I can absolutely categorically tell you it never happened."
"I have no recollection of ever meeting this lady, none whatsoever."
He said Ms Giuffre's account of him "profusely sweating" and "pouring with perspiration" when they danced at the club on the night in 2001 when she says they first had sex was impossible, because he had a medical condition preventing him from perspiring.
Addressing Ms Giuffre's claims that she had dined with the prince, danced with him at a nightclub, and went on to have sex with him at the house of Ghislaine Maxwell, a friend of the prince, in Belgravia, central London, he said "there are a number of things that are wrong with that story".