Answering questions about his links to convicted sex offender Jeffrey Epstein Prince Andrew said the alleged incidents “never happened”.
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 an American financier who, at the age of 66, 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,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.
He had investigations carried out to establish whether a photograph of him with Ms Giuffre was faked, but they were inconclusive.
He would testify under oath if “push came to shove” and his lawyers advised him to.
He was unaware of an arrest warrant against Epstein when he invited him to Princess Beatrice’s 18th birthday party at Windsor Castle.
He does not regret his friendship with Epstein because of “the opportunities I was given to learn” from him about trade and business.
Speaking out about his relationship with the financier had become almost “a mental health issue” for him.
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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”.
He said the date when Ms Giuffre says he had sex with her was 10 March 2001, when he had taken his daughter Beatrice to Pizza Express in Woking for a party before spending the night at home.
“Going to Pizza Express in Woking is an unusual thing for me to do,” he said. “I remember it weirdly distinctly.
Ms Giuffre described him providing her with alcohol at a nightclub, but Prince Andrew said: “I don’t drink, I don’t think I’ve ever bought a drink in Tramps whenever I was there.
On claims he was sweating, he said: “I have a peculiar medical condition which is that I don’t sweat or I didn’t sweat at the time,” he said, blaming it on “an overdose of adrenaline in the Falklands War”.
He said he had only started to be able to sweat again “in the recent past”.
Asked about a photograph of him and Ms Giuffre being taken at Ghislaine Maxwell’s house, he said he had “absolutely no memory” of it.
British Prince denies Virginia Giuffre allegation,who says she was forced to have sex with him aged 17.