The actor, then aged 11, signed the hardback during filming of Harry Potter and the Philosopher's Stone in North Yorkshire in 2000.

He told the book's owner - an extra in the film - she was the first person to ask for his signature.

The book, which was also signed by Emma Watson and Rupert Grint, was auctioned by Teesside-based Vectis.

Vectis said the book's owner, Verity Collins, approached Radcliffe as he sat alone in a marquee on set at Goathland on what is believed to have been the first day of filming.

The North Yorkshire village was shown on screen as the fictional Hogsmeade.

He wrote: "To Verity, best wishes Daniel Radcliffe."

Robbie Coltrane, who played Hagrid in the franchise, was another star to give his autograph while the film's director, Chris Columbus, wrote "Believe in magic" as Ms Collins was having chemotherapy to treat leukaemia.

Kathy Taylor, from Vectis, said the book had smashed its "conservative" £500 estimate.

"It's such a lovely story. It's always difficult to gauge the price when something is unique," she said.

"The seller, who now lives in the south, has great memories of her time on the film and the way the cast treated her during a difficult time while she was ill but her mum said recently it had been sitting in a drawer.

"It became apparent when we promoted the sale on social media it was likely to have been signed on the first day of filming on what was the first Harry Potter film, with actor Chris Rankin, who played Percy Weasley, confirming he had signed the book as well."

Sold alongside the book were photographs of Ms Collins wearing a Hogwarts school uniform, cast documents and her invite to Leavesden Studios for a tour of sets used in the film.

In addition to the hammer price of £2,600, the buyer will pay an additional 25% commission - taking the total to £3,250.