It was sold at Cheffins in Cambridge to a private collector who bid for it by telephone, with the letter fetching more than 11 times its pre-sale estimate of £500 to £1,000.
The letter was first sold in 1935 at Maggs Bros, one of the world's largest and oldest antiquarian booksellers, for £38.
Charles Ashton, a director at Cheffins said: "This letter is a defining moment of history showing the King's intention to go to war with France and Napoleon."
The King wrote in the letter that he had "perused the dispatch and private letter from Lord Whitworth", a British politician and diplomat who at the time of the letter was ambassador in Paris.
"The conduct of France has been equally unfair to the last," he wrote.
"And though conscious of the Evils that must be entailed on many Countries by the renewal of War, yet the conviction that by the restless disposition of the Ruler of France this event could not long have been kept off, it seems necessary to attend alone to the best modes of repelling the violence with effect, and the attacking those objects which our present means render attainable.
"The King will remain in Town to execute any Steps that the present moment may require."