Gary McKinnon was arrested in June following charges by US prosecutors that he illegally accessed government computers - including systems at the Pentagon, army, navy and Nasa - causing $700,000 worth of damage.
McKinnon could face up to 70 years in jail and fines of up to $1.75 million.
McKinnon, whose hacking name was Solo, said he was planning to appeal against the decision. "I am very worried and feeling very let down by my own government," he said.
He had tried to fight extradition, saying he was "already hung and quartered over there" and would not receive a fair trial.
McKinnon, an unemployed computer system administrator who lives in London, denied causing any damage and said he was looking for evidence that America is concealing the existence of UFOs.
Prosecutors said McKinnon hacked into sensitive networks over a one year period from February 2002, crippling US defence systems in the wake of the September 11, 2001 attacks.
John Reid, the home office minister, signed the order for his extradition, his ministry said.
A judge ruled in May that McKinnon, who has been indicted in New Jersey and northern Virginia, should be sent to the US to face trial. The decision required Reid's approval.
McKinnon has 14 days to appeal against the decision.