Dated 12 August, 1960, it was discovered inside a book bought at a car boot sale in Liverpool by a coin collector.
Nothing is known about who the letter was addressed to or if they were auditioned.
A telephone bidder from Europe bought it at Christie's auction house.
The letter, which was written by a young Sir Paul in response to a notice in the Liverpool Echo that read "Drummer - Young - Free", had been expected to fetch no more than £9,000.
The band, which included bass player Stuart Sutcliffe, was looking for a drummer for their planned trip to Hamburg.
Three days after the letter's date they travelled to Germany with their new drummer Pete Best.
Until the discovery of the letter it was believed that Best had been the only drummer who the group had considered.
Best was replaced by Ringo Starr two years later.
Neil Roberts, of Christie's, said: "One of the best aspects of my work is the rare occasion when, out of the blue, you are made aware of the existence of something so extraordinary, it alters the knowledge of your specialist field. This letter has proved to be such a case.
"It is exciting to be able to offer to market a newly discovered important item of Beatles memorabilia, on behalf of an individual who was fortunate enough to find it folded up in a book at a car boot sale."
Another item that sold for £97,250 at the rock and pop memorabilia auction was a hand-written sign that John Lennon and Yoko Ono displayed during one of their "bed-ins" at Queen Elizabeth hotel in Montreal in 1969.Last Updated: 11/18/11 09:57