His wife then called for the case against her husband, who has been on remand in jail for five years, to be dropped.
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Assange, 52, is accused of conspiracy to obtain and disclose national defence information following the leaking of hundreds of thousands of documents relating to the Afghanistan and Iraq wars.
At a hearing in February, his lawyers asked to challenge a previous dismissal of a case to stop extradition.
A final High Court decision was deferred until yesterday, after the US promised he will not face the death penalty and can rely on the right to free speech at any trial.
Supporters cheered outside the London court as news filtered out that he can appeal.
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Assange’s wife Stella hailed “the right decision” then called on the US to drop the “shameful” case.
The mother of two added: “How long can this go on for?
“Our eldest son just turned seven.
“All their memories of their father are in the visiting hall of Belmarsh.”
If successful, Assange may avoid extradition, though a US appeal would be likely.