Any hope the Duke may have had that he could put the allegations behind him with a speech at the World Economic Forum were dashed as his accuser, Virginia Roberts, made a series of lurid claims in her first sworn court statement.

She claimed the Duke had a “sexual interest in feet”, that she had sex with him while his police protection officers sat in a car outside and that his friend Jeffrey Epstein, a convicted sex offender, was “collecting private information” about the Queen’s son.

The Duke has categorically denied any form of sexual contact with Miss Roberts, now 31. She described the denial as “hurtful”.

The Duke, who will make a brief on-camera reference to Miss Roberts’s claims when he addresses an event for entrepreneurs in Davos, Switzerland this evening, refused to comment on the latest developments as he left a chalet in the nearby resort of Klosters this morning.

He later arrived at the conference in Davos where he did his best to carry on with business as usual by holding a series of one-on-one meetings promoting British entrepreneurs, education and science and technology.

Miss Roberts, who now lives in Colorado with her husband and two children, claims to have had sex with the Duke on three occasions in 2001, when she was 17.

On January 19 she signed a sworn statement – the first time she has made the accusations about the Duke under oath – which was submitted to a court in Florida as part of an attempt to have a plea bargain agreement with Epstein overturned.

“I did have sexual contact with him as I have described here – under oath. Given what he knows and has seen, I was hoping that he would simply voluntarily tell the truth about everything. I hope my attorneys can interview Prince Andrew under oath about the contacts and that he will tell the truth.”

In a separate development, lawyers have written to the Duke at Buckingham Palace to ask him to testify under oath about Miss Roberts’s claims. According to one report, the Palace refused to accept delivery of the letter.

Miss Roberts says she first had sex with him at the London home of Epstein’s girlfriend Ghislaine Maxwell.

“I knew he was a member of the British Royal Family, but I just called him ‘Andy’,” she said. “I got news from Maxwell that I would be meeting a prince. Later that day, Epstein told me I was meeting a ‘major prince’. Epstein told me ‘to exceed’ everything I had been taught. He emphasized that whatever Prince Andrew wanted, I was to make sure he got.

“Eventually Prince Andrew arrived, along with his security guards. The guards then went out of the house and stayed out front in their car. It was just Epstein, Maxwell, and me inside alone with Andy. I was introduced to the Prince, and we kissed formally, cheek to cheek.”

They went for dinner and to Tramps nightclub, where “Epstein and Maxwell informed me that the Prince wanted to see ‘more of me’,” then returned to the flat where she “engaged in sexual activities” with the Duke in a bedroom and a bathroom.

“I chatted with Epstein about this the next day. I told him, ‘it went great’. Epstein said something to the effect of, ‘You did well. The Prince had fun.’ I felt like I was being graded.

“It was horrible to have to recount all these events and have to try to meet all these needs and wants. I told Epstein about Andy’s sexual interests in feet. Epstein thought it was very funny. Epstein appeared to be collecting private information about Andy.”

She claims she Epstein paid her $15,000 (£10,000) and that: “That money was for what I had done and to keep my mouth shut about ‘working’ with the Prince.”

Miss Roberts alleges that the second time she had sex with the Duke was in Epstein’s New York mansion in spring 2001.

Epstein and Maxwell were making lewd jokes about “Randy Andy” and, she says: “I had the impression that Andy had come there to see Epstein and to have sex with me. There was no other apparent purpose for Andy to be there.”

She went upstairs to a massage room, which she referred to as “the dungeon” and was paid $400 by Epstein “for servicing Andy that time”.

She adds: “The third time I had sex with Andy was in an orgy on Epstein’s private island in the U.S. Virgin Islands. I was around 18 at the time. Epstein, Andy, approximately eight other young girls, and I had sex together. The other girls all seemed and appeared to be under the age of 18 and didn’t really speak English.

“Epstein laughed about the fact they couldn’t really communicate, saying that they are the ‘easiest’ girls to get along with.” Afterwards she “felt disgusted”.

She signed the statement saying: “I declare under penalty of perjury that the foregoing is true and correct.”