They will hold another hearing later this month to determine whether he can appeal.
"I have not been charged with any crime in any country," he said on the steps of the High Court in London. "Despite this, the European arrest warrant is so restrictive that it prevents UK courts from considering the facts of a case, as judges have made clear here today."
Assange is accused of sexually assaulting two women in Sweden in August 2010. Although he has not been charged with a crime, Swedish prosecutors want to question him in connection with the allegations.
The court comprehensively rejected his defense against being sent there to face prosecution, and was particularly scathing about a dispute with one of the women over whether she had consented to having sex with him.
Swedish authorities allege that the unnamed woman agreed to have sex with him only if he wore a condom, and that he then had unprotected sex with her while she was asleep.
"The allegation is that he had sexual intercourse with her when she was not in a position to consent and so he could not have had any reasonable belief that she did," the court said.
Assange drew cheers from the crowd as he left the court. A "Free Assange" rally was planned for Wednesday outside the Royal Courts of Justice.
Assange, an Australian, decided to fight the case at the High Court after a judge at Belmarsh Magistrates' Court ruled in February that the WikiLeaks head should be extradited.
Assange denies the accusations, saying they are an attempt to smear him, and he says it would be unfair to send him to a country where the language and legal system are alien to him. His attorneys have fought his extradition on procedural and human-rights grounds.
Assange's lawyers have suggested that Sweden would hand him over to the United States if Britain extradites him. The prosecutor representing Sweden has dismissed that claim.
The extradition case is not linked to his work as founder and editor-in-chief of WikiLeaks, which has put him on the wrong side of the U.S. authorities.
His organization, which facilitates the anonymous leaking of secret information, has published some 250,000 confidential U.S. diplomatic cables in the past year, causing embarrassment to the government and others.
It has also published hundreds of thousands of classified U.S. documents relating to the conflicts in Iraq and Afghanistan.
But the organization has come under increasing financial pressure in recent months, leading Assange to announce last week that WikiLeaks was temporarily stopping publication to "aggressively fundraise" in order to stay afloat.
A financial blockade by Bank of America, VISA, MasterCard, PayPal and Western Union has destroyed 95% of WikiLeaks' revenue, Assange said.
Many financial institutions stopped doing business with the site after it released the U.S. diplomatic cables late last year, and donations have been stymied.
U.S. authorities have said disclosing the classified information was illegal and caused risks to individuals and national security.
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