From BBC News:
A petition on the No 10 website had called for a posthumous government apology to the computer pioneer.
In 1952 Turing was prosecuted for gross indecency after admitting a sexual relationship with a man. Two years later he killed himself.
The campaign was the idea of computer scientist John Graham-Cumming.
He was seeking an apology for the way the mathematician was treated after his conviction. He also wrote to the Queen to ask for Turing to be awarded a posthumous knighthood.
The campaign was backed by author Ian McEwan, scientist Richard Dawkins and gay-rights campaigner Peter Tatchell. The petition posted on the Downing Street website attracted thousands of signatures.
Mr Brown, writing in the Telegraph newspaper, said: “While Mr Turing was dealt with under the law of the time and we can’t put the clock back, his treatment was of course utterly unfair and I am pleased to have the chance to say how deeply sorry I and we all are for what happened to him.”