Julian Assange, 40, fights against extradition to Sweden to face accusations of sex offences.
Lord Phillips, the court’s president said he had lost by a majority of five justices to two.
The court ruled the extradition request had been “lawfully made”.
However, Julian Assange’s lawyers have been given 14 days to consider challenging the ruling, saying it could have been reached unfairly.
Dinah Rose QC, for Julian Assange, said the Supreme Court’s decision could have been made on legal points not argued during the appeal – and she needed time to consider asking the court to reopen the case.
Julian Assange, who has been on conditional bail in the UK, did not attend the hearing in central London. His lawyer later told reporters he had been “stuck in traffic”.
The Australian is accused of raping one woman and “sexually molesting and coercing” another in Stockholm in August 2010, but he claims the allegations against him are politically motivated.
Julian Assange’s lawyers had asked the court to block his removal, arguing that a European arrest warrant issued against him was “invalid and unenforceable”.
The key legal question for the seven judges was whether the prosecutor who issued the arrest warrant had the “judicial authority” to do so under the 2003 Extradition Act.
Lord Phillips said five of the justices agreed the warrant had been lawful because the Swedish prosecutor behind the warrant could be considered a proper “judicial authority” even it they were not specifically mentioned in legislation or international agreements.
This point of law had not been simple to resolve, said Lord Phillips, and two of the justices, Lady Hale and Lord Mance, had disagreed with the decision.
But Dinah Rose immediately indicated she could challenge the judgement saying that it relied on a 1969 convention relating to how treaties should be implemented. She said this convention had not been raised during the hearing.
In a statement, the Supreme Court said: “Following this morning’s judgment by the Supreme Court of the United Kingdom in Assange v The Swedish Prosecution Authority, Ms Rose has indicated that she may make an application to re-open the court’s decision.
“Ms Rose suggested that the majority of the court appear to have based their decision on the interpretation of the Vienna Convention on the Law of Treaties, on which no argument was heard and no opportunity of making submission was given.
“The Supreme Court has granted Ms Rose 14 days to make such an application. If she decides to do so, the justices will then decide whether to re-open the appeal and accept further submissions either verbally through a further hearing, or on paper on the matter.”
The decision to stay the extradition order means that it cannot become active until 13 June – but it would be further delayed were there to be additional submissions.
Julian Assange’s Wikileaks website published material from leaked diplomatic cables embarrassing several governments.
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