"We hope this situation will come to an end very soon and we look forward to the decision next week", Robinson said.
Mr Assange tweeted: "London case is now becoming very interesting as we expose the improper conduct of CPS and the U.S. case".
According to Assange, only the first technical point of his legal challenge has failed.
The Judicial Office tweeted the judgment on Tuesday afternoon.
"Having considered the application set out I am not persuaded that the warrant should be withdrawn", she said.
If the British case against Assange is dropped, it might make it easier for Assange to obtain diplomatic status, which could ease his way for possible travel to Ecuador without risking arrest and extradition proceedings.More news: Red Dead Redemption 2 info leak reveals plenty about game
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Sweden dropped its investigation previous year, but British police are still seeking to arrest Assange for failing to surrender to a court after violating his bail terms during his unsuccessful battle against extradition.
His lawyers had argued that with the Swedish case dropped, there was no longer any justification for the arrest warrant against him, but judge Emma Arbuthnot rejected their reasoning. His defense papers asserted that he has been living in conditions "akin to imprisonment" and that his "physical and psychological health have deteriorated and are in serious peril".
He added: "The proper approach is that when a discrete, standalone offence of failing to surrender occurs, it always remains open to this court to secure the arrest". "Therefore a warrant stands", Watkins said.
The country has already given him citizenship.
The WikiLeaks founder fears that Sweden would extradite him to the United States, where he faces prosecution for WikiLeaks' publication of leaked USA military and diplomatic documents.
If arrested he would face rendition to the US, treatment similar to that meted out against Wikileaks whistleblower Chelsea Manning - and possible "persecution, indefinite solitary confinement and the death penalty", Summers said in a written submission.
The Australian was being investigated for rape in Sweden, which he has denied.