Lauri Love, 31, faces a lengthy prison term if convicted of charges including hacking into the networks of the US Federal Reserve, US Army and NASA, among others in 2012 and 2013.
Love's supporters said he would appeal the decision.
US prosecutors argue that Love disseminated the personal information of US citizens including serving members of the military.
District judge Nina Tempia granted the US's extradition request in a hearing in London but also gave Love leave to appeal the decision to a higher court.
Love argues he should face justice in Britain rather than the US and believes that his Asperger Syndrome condition would get worse if he was sent to an American prison.
"If you have come for justice then you have missed it," Love told journalists after the decision.
"We shouldn't look on this as even nearly the end... We will continue to fight because we're doing what's right," he added outside court.
She added: "I know that Lauri's legal team will apply to appeal this ruling and the Courage Foundation will continue to support Lauri until his safety is assured.
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