Two judges at the High Court in London rejected arguments by the 40-year-old Australian, whose anti-secrecy website has enraged governments around the world, that his extradition would be unlawful.
"The court dismissed the appeal," said a summary of the judgement, before detailing the four counts on which Assange had appealed against a decision by a lower court in February that he should be sent to Sweden.
Assange said he would consult his lawyers about whether to make a further appeal to England's Supreme Court, but doing so would be difficult as judges must first decide that the case is of special public interest.
"We will be considering our next steps in the days ahead," the former computer hacker told a scrum of reporters and cameramen gathered from around the world, in a brief statement from the steps of the court.
Assange has strongly denied the allegations, claiming they are politically motivated and linked to the activities of WikiLeaks. He has been under virtual house arrest since he was first detained in December.
He wore a blue suit with a poppy in his lapel -- a symbol in Britain to mark Armistice Day on November 11, the anniversary of the end of World War I -- and sat quietly next to his lawyers as the judgement was announced.
During the appeal hearing in July, Assange had argued that the European Arrest Warrant under which he was held last December was invalid because it was issued by a prosecutor and not a court.
But the British judges said it had been subjected to proper judicial scrutiny in Sweden.
They also rejected his assertion that the claims made by two women of unlawful coercion, two counts of sexual molestation and an accusation of rape would not be offences under English law.
One woman alleged that Assange had unprotected sex with her while she was asleep, and the judges rejected his lawyers' contention that consent to sex with a condom remained consent when a condom was not used.
"There is nothing in the statement from which it could be inferred that he reasonably expected that she would have consented to sex without a condom," the court ruling said.
The judges also rejected Assange's argument that he should not be extradited because he was only wanted for questioning and had not been charged, saying he was "plainly accused" of the crimes.
The fourth and final ground of appeal was that the arrest warrant was disproportionate, given that Assange had offered to be questioned via videolink, but the court also dismissed this.
The enigmatic WikiLeaks boss has been living under strict bail conditions, including having to wear an electronic ankle tag and observe a strict curfew, at the east England mansion of supporter and former army captain Vaughan Smith.
Smith was in court on Wednesday along with supporters including campaigning journalist John Pilger.
Assange now has 14 days to decide whether he will try to take the case to the Supreme Court of England and Wales, the highest legal authority in the land.
But leave to appeal can only be granted by either the High Court or the Supreme Court, and then only if it there is a point of law of general public importance.
In an autobiography published in September, Assange repeated his denial of the rape allegations.
"I did not rape those women and cannot imagine anything that happened between us that would make them think so, except malice after the fact, a joint plan to entrap me, or a terrifying misunderstanding that was stoked up between them," he wrote.
"I may be a chauvinistic pig of some sort but I am no rapist, and only a distorted version of sexual politics could attempt to turn me into one."