The seven-judge panel ruled five to two that Mr Assange's extradition appeal be dismissed and he be extradited.
The panel decided Sweden's prosecutor was a judicial authority and therefore the European arrest warrant issued for Mr Assange was valid.
"The request for Mr Assange's extradition has been lawfully made and his appeal against extradition is accordingly dismissed," Supreme Court president Judge Nicholas Phillips said.
Swedish authorities argued before the hearing that if Mr Assange's appeal was granted, it could throw the fast-tracked European arrest warrant system into turmoil, with implications across the continent.
A Swedish prosecutor wants to question Mr Assange over allegations he sexually assaulted two women in August 2010.
Mr Assange has previously said the sex was consensual, and that the allegations are politically motivated.
The claims followed Mr Assange's release of a swathe of leaked United States diplomatic cables that embarrassed governments and international businesses.
Mr Assange fears he may be extradited to the US from Sweden.
Mr Assange was not present in the court to hear the verdict.
One of his supporters, journalist and filmmaker John Pilger, said he was "stuck in traffic" with his mother Christine, who flew in from Australia for the verdict.
Christine Assange said ahead of the judgement: "It's a 24-hour nightmare because we know he is not safe and the biggest governments in the world are gunning for him."
Mr Assange's defence team has been granted a two-week stay to decide whether to appeal against a particular point of law.
If this is denied, Mr Assange's last lifeline will be the European Court of Human Rights.
One of his lawyers, Jennifer Robinson, told the ABC the Supreme Court's decision was "not altogether unexpected".
Ms Robinson said "it remains to be seen" what the next options are.
She also said she had received a letter from Attorney-General Nicola Roxon, who told her she had no knowledge about US plans to charge Mr Assange.
US ambassador to Australia Jeffrey Bleich said: "[There's] absolutely no basis for the US to be interested in this, we're not involved".
Greens Senator Scott Ludlam said the Australian Government could be doing more to help Mr Assange.
"I think the Australian Government has been extremely weak and they need to decide whether their allegiances are with Washington or the Australian public, which is, as someone has already said, actually very strongly in support of Julian and his colleagues," he said.
The Swedish lawyer for Mr Assange's alleged victims said he expected Mr Assange will be extradited to Sweden sooner or later.
"[Today's] decision was what we expected," lawyer Claes Borgstroem said.
"It's unfortunate that it has been delayed further, but he will ultimately be extradited.
"We expect an indictment fairly soon after he gets here, maybe within a month."
One week ago, Mr Assange attended a film screening in London wearing a Guy Fawkes mask.
"This may be my last time in public, so I thought I should start with a situation where you won't be able to see me anymore," he said.
Since December 2010, Mr Assange has been forced to report to police daily and wear an electronic ankle tag.
A lower court in Britain initially approved Mr Assange's extradition to Sweden in February 2011.
An appeal to the High Court was rejected in November, but Mr Assange subsequently won permission to appeal to the Supreme Court, which heard two days of complex arguments in February.