In an order on Friday, judges said that then-home secretary Priti Patel's decision to order Assange's extradition to the US in June 2022 had been overturned.
It comes after the WikiLeaks founder flew into Australia on Wednesday following his release from a London prison and an appearance before a judge in the US territory of the Northern Mariana Islands.
Assange pleaded guilty to a single charge on Tuesday after the US dropped 17 other espionage charges against him, before arriving in Australia.
The 52-year-old had been due to bring a challenge at the High Court in London in early July over his extradition.
However, Friday's order from justices Victoria Sharp and Adam Johnson said the legal proceedings had been formally discontinued.
Assange had been locked in a lengthy legal battle to avoid being extradited to the US, which saw him live in the Ecuadorian embassy in London in 2012 before his detention in Belmarsh.
In a January 2021 ruling, then-district judge Vanessa Baraitser said Assange should not be sent to the US, citing a real and "oppressive" risk of suicide, while ruling against him on all other issues.
Later that year, US authorities won a High Court bid to overturn this block, paving the way towards his extradition.
Assange's planned challenge was over the original judge's dismissal of parts of his case, as well as Patel's decision.
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