Mr Hague was speaking after the Wikileaks founder was granted political asylum by Ecuador in a move that escalated the tension in a diplomatic stand-off between the South American country and the UK.
While Assange remained holed up in the Ecuadorian embassy in London last night, Mr Hague held a press conference to issue his uncompromising warning.
Mr Hague said it was a “matter of regret” that the Ecuadorian government decided to grant the Wikileaks founder political asylum, but warned that it “does not change the fundamentals” of the case.
Assange has been sheltering in the embassy in London’s Knightsbridge since June to avoid extradition to Sweden, where he faces questioning over assault and rape claims – allegations that he denies.
The allegations were levelled against Assange by two women he met while on a trip to Sweden in the wake of some of his organisation’s controversial
disclosures of US intelligence material.
The women accuse him in separate cases of molestation and rape, and Swedish authorities have been seeking his extradition since late 2010.
Assange has expressed fears that the case is the opening gambit in a Washington-orchestrated bid to make him stand trial for his leaks in the US – something disputed both by Swedish authorities and the women involved. Assange fears he could face the death penalty if he is found guilty in America.
Granting Assange asylum, the Ecuadorian foreign minister Ricardo Patino said it was clear that if Assange were extradited to the US “he would not have a fair trial, could be judged by special or military courts and it’s not implausible that cruel and degrading treatment could be applied, that he could be condemned to life in prison or the death penalty”.
Mr Patino added that there were “serious indications” that the US could threaten Assange’s “security, integrity and even his life”.
Ecuador took the step of granting the Wikileaks founder asylum after Ecuadorian ministers accused the UK of threatening to seize Assange from inside the embassy.
Under international law – the 1961 Vienna Convention on Consular Relations – diplomatic posts are considered the territory of the foreign nation.
But, in a letter to Ecuadorian officials, Britain cited a little-known law, the Diplomatic and Consular Premises Act of 1987, which it suggested would allow it to arrest Assange within the embassy premises.
The law gives Britain the power to revoke the status of a diplomatic mission if the state in question “ceases to use land for the purposes of its mission or exclusively for the purposes of a consular post”, but only if such a move is “permissible under international law”.
In its letter, British authorities added: “We very much hope not to get to this point.”
Mr Hague dismissed Ecuadorian claims that they had been threatened with an “attack” on their embassy. “There is no threat here to storm an embassy. We are talking about an act of parliament in this country which stresses that it must be used in full conformity with international law,” he said.
The Foreign Secretary denied claims by Assange that there was a deal which would see him extradited to the US.
He said: “We have no arrangement with the United States. This is the United Kingdom fulfilling its obligations under the Extradition Act to Sweden, a close partner in so many ways, a fellow democracy in the European Union.
“It is as simple as that. Therefore to us it is a simple matter of carrying out our law, but as well as being simple it is something we must do. We absolutely must fulfil our obligations under the Extradition Act.
“Therefore we are determined to do so and we remain determined to do so despite the regrettable announcement that Ecuador has made today.”