The 40-year-old Australian will take his legal battle to the High Court in London for a two-day hearing, in the hope of reversing the February ruling.
A judge rejected arguments by Assange's defence team that he would face an unfair trial in Sweden that would breach his human rights.
He was arrested in December after two Swedish women accused him of sexual assault, allegations that Assange denies, as his whistleblowing website was in the process of releasing a huge cache of leaked US diplomatic cables.
It was the site's latest dump of American government documents and infuriated Washington.
Swedish authorities want to quiz him over the sex assault claims, although he has not been formally charged.
Scores of journalists attended his previous court hearings as well as celebrity supporters including socialite Jemima Khan and human rights campaigner Bianca Jagger.
After judge Howard Riddle rejected his lawyers' arguments during the extradition hearing at Belmarsh Magistrates' Court in London, Assange lashed out at the decision and blamed the European warrant system under which he was arrested.
"It is a result of the European Arrest Warrant system run amok. There was no consideration during this entire process as to the merits of the allegations against me," he told reporters after the ruling.
Even if he loses this week's appeal, Assange has signalled he is prepared for a lengthy legal battle and could take his challenge all the way to the Supreme Court.
The former computer hacker has been living under strict bail conditions, including wearing an electronic ankle tag and a curfew, at a friend's mansion in eastern England since December.
Despite the restrictions, on Sunday he managed to celebrate his 40th birthday, which was on July 3rd. He held a party to which more than 100 guests were invited, according to Vaughan Smith, who owns the country estate where Assange is staying.
He has claimed his greatest fear is eventual extradition to the United States, where his lawyers argued he could be sent to the Guantanamo Bay detention facility or face the death penalty.
US authorities opened a criminal investigation against Assange in July 2010 but are yet to formally bring any charges against him.
WikiLeaks infuriated Washington with its release of classified documents, including around 250,000 US diplomatic cables and thousands of secret files about the wars in Afghanistan and Iraq.
American soldier Bradley Manning has been accused of leaking documents to the whistleblower site and is in a US military prison awaiting a possible court martial on charges that include "aiding the enemy."
US Vice President Jo Biden has slammed Assange as a "hi-tech terrorist", but he has also won many admirers -- his website is among the nominees for this year's Nobel Peace Prize, due to be announced in October.