Britain's highest court is Assange's final avenue of appeal under UK law, having been detained in December 2010 on a European arrest warrant. He is wanted in Sweden for questioning over allegations of rape and sexual assault.
Since then, the 40-year-old Australian has been through round upon round of legal battles, culminating in what will be a short ruling at the Supreme Court in central London.
The judgment, expected to take around 10 minutes, will be handed down at 9:15am (0815 GMT) Wednesday, streamed on the Sky News website and published online once delivered.
Assange will have been living under restrictions on his movement for 540 days when the verdict is handed down.
Assange's case rests on a single point -- that the Swedish prosecutor who issued a warrant for his arrest was not a valid judicial authority.
The Supreme Court president will give a summary of the point of law raised by the appeal, the court's decision, and a brief explanation of its rationale.
A lower court in Britain initially approved Assange's extradition to Sweden in February 2011. An appeal to the High Court was rejected in November, but he subsequently won permission to appeal to the Supreme Court.
If that court rejects his appeal, Assange will have exhausted all his options in Britain but he could still make a last-ditch appeal to the European Court of Human Rights in Strasbourg, France.
Assange has said he fears his extradition would eventually lead to his transfer to the United States, where US soldier Bradley Manning is facing a court-martial over accusations that he handed documents to the anti-secrecy website WikiLeaks.
On May 23, Assange attended a screening in London wearing a Kevlar 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 explained.
Assange added: "I think all of us are at our best when we are pursuing an ideal that we find to be important to ourselves and important to others.
"I feel that I have made my days count, so I certainly would not want to exchange days that can be counted for days that cannot."
The Supreme Court is the final court of appeal in Britain for civil cases. It hears appeals in criminal cases from England, Wales and Northern Ireland. It also hears cases of the greatest public or constitutional importance affecting the whole population.
The court sits on Parliament Square in central London, directly opposite the Houses of Parliament and with the Treasury and Westminster Abbey on either side.
The judgment will be handed down in the building's largest courtroom.
WikiLeaks supporters are expected to muster in numbers for the verdict.
Would-be attendees were pre-warned of airport-style security and that items of clothing or other materials with messages that "undermine the dignity of the court or which seek to interfere with the proper administration of justice" will not be allowed inside.
In February the Supreme Court heard two days of complex arguments from lawyers for both sides.
The legal saga began in August 2010 when Assange was accused of raping one woman by having sex with her while she was asleep, and of sexually assaulting another woman, in Sweden.
The former computer hacker insists the sex was consensual and says the allegations are politically motivated.
Assange shot to fame earlier that same year when WikiLeaks enraged Washington by leaking thousands of secret US documents on the wars in Iraq and Afghanistan.
After he was arrested in London in December 2010 he spent more than a week in jail before being freed to live under strict bail conditions amounting to virtual house arrest.