Lord Phillips, the court's president said he had lost by a majority of five justices to two.
The court ruled the extradition request had been "lawfully made".
However, Mr Assange's lawyers have been given 14 days to consider challenging the ruling, saying it could have been reached unfairly.
Dinah Rose QC, for Mr Assange, said the Supreme Court's decision could have been made on legal points not argued during the appeal - and she needed time to consider asking the court to reopen the case.
Mr Assange, who has been on conditional bail in the UK, did not attend the hearing in central London. His lawyer later told reporters he had been "stuck in traffic".