But a question mark hung over the decision after Assange's lawyer made the highly unusual suggestion that she would try to reopen the case, raising the prospect of more legal wrangling.
Assange, 40, has spent the better part of two years fighting attempts to send him to Sweden, where he is wanted over sex crime allegations. He has yet to be charged.
The U.K. side of that struggle came to an uncertain end Wednesday, with the nation's highest court ruling 5-2 that the warrant seeking his arrest was properly issued - and Assange's lawyer saying she might contest the ruling.
Supreme Court President Nicholas Phillips, reading out the verdict, acknowledged that coming to a conclusion in the high-profile case had "not been simple."
Assange's story has been shot through with international intrigue, online activism, and scandal. But the case before the Supreme Court hinged on a narrow technicality: Did Swedish officials properly order his arrest?
His lawyers say "no." A prosecutor, not a judge, issued the warrant, a practice they've described as arbitrary and unfair. Swedish officials say "yes," arguing that, in Sweden as in other European countries, prosecutors carry out a quasi-judicial function.
The Supreme Court came down on Sweden's side Wednesday, with Phillips ruling that "the request for Mr. Assange's extradition has been lawfully made and his appeal against extradition is accordingly dismissed."
But scarcely had Phillips finished speaking before Assange lawyer Dinah Rose was on her feet, complaining that the court's ruling largely relied on a treaty whose interpretation she says she never had the chance to challenge.
In a surprise move, she requested extra time to study the judgment with an eye toward trying to reopen the case.
Phillips gave Rose two weeks to present her case, meaning an extradition wouldn't happen until the second half of June at the earliest.
Such a maneuver is practically unheard of, according to attorney Karen Todner, whose law firm handles many high-profile extradition cases.
"It's very unusual," she told The Associated Press. "I've never known them to reopen a case."
Any eventual extradition of Assange could, however, be much later. Even if the Supreme Court refuses to revisit its judgment, Assange could appeal to the European Court of Human Rights, although Todner said he was unlikely to make much headway there unless he could argue that his physical safety or psychological well-being would be at risk in Sweden.
Assange, a former computer hacker from Australia, shot to international prominence in 2010 with the release of hundreds of thousands of secret U.S. documents, including a hard-to-watch video that showed U.S. forces gunning down a crowd of Iraqi civilians and journalists that they'd mistaken for insurgents.
His release of a quarter-million classified U.S. State Department cables in the final months of that year outraged Washington and destabilized American diplomacy worldwide.
But his work exposing government secrets increasingly came under a cloud after two Swedish women accused him of molestation and rape following a visit to the country in mid-2010. Assange denies wrongdoing, saying the sex was consensual, but has refused to go to Sweden, claiming he won't get a fair trial there.
He and his supporters have also hinted that the sex allegations are a cover for a planned move to extradite him to the United States, where he claims he's been secretly indicted for the WikiLeaks disclosures.
Those allegations, paired with the ponderous progress of Assange's appeals, have caused irritation in Sweden.
Claes Borgstrom, the lawyer who represents the two Swedish women who accuse Assange of sex crimes, expressed support for the verdict, but said he found it "difficult to understand" why the defense had been given the opportunity to try to reopen the case.
He dismissed suggestions that the underlying motive behind the extradition was to hand Assange over to the U.S.
"He is not at a greater risk of being handed over from Sweden than from Britain," he told the AP.
Borgstrom wasn't the only one expressing impatience. Ola Lofgren, of the Swedish prosecutor general's office, said it was in everyone's interest that the judicial back-and-forth "is shortened as much as possible."
Australia's government also weighed in, saying in a statement that it would "closely monitor" any proceedings against Assange in Sweden.
Unusually, Assange did not appear in court Wednesday; he was reportedly stuck in traffic. The WikiLeaks chief has spent much of the past 18 months living in a supporter's mansion in rural England.
Although his website has languished amid legal and financial issues, he's moved on to other projects, including a WikiLeaks-themed social networking site and a talk show on the Russian government's English-language broadcaster, RT.
Attempts to reach him for comment on the verdict weren't immediately successful.
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Karl Ritter and David MacDougall in Stockholm and Rod McGuirk in Sydney contributed to this report.
