A US court indicted McKinnon, 42, in November 2002, when it was alleged that he accessed more than 90 computers belonging to NASA and the Pentagon, causing damage costing around $900,000 (£450,000).
The US government says he committed a malicious crime - the biggest military computer hack in history.
McKinnon, a self-proclaimed ‘computer nerd' who claimed he was searching for information on UFOs, could now face decades in jail in the US – and possibly even a life sentence.
In 2006, British courts ruled that McKinnon should face trial in the US. He unsuccessfully appealed to the High Court, which was then followed by this latest case before the House of Lords.
His family and campaigners had lobbied for him to be tried in the UK – and asked that he be allowed to serve any sentence here if he is found guilty.
McKinnon was caught in 2002 when some of the software used was traced back to his girlfriend's email account.