Radical cleric Abu Hamza and four other terrorist suspects wanted by the US authorities will be handed over "as quickly as possible", Home Secretary Theresa May has said.
She was speaking after human rights judges ruled there would be no violation of the European Human Rights Convention if the UK extradited the five to the US to face a range of terrorist charges.
The men had claimed the prospect of solitary confinement in a "supermax" high-security jail and sentences of life imprisonment without parole would breach a European ban on "torture or inhuman or degrading treatment or punishment".
The ruling, after a series of controversial human rights judgments against the Government, was welcomed by Prime Minister David Cameron. During a trade mission to south-east Asia, he said: "It is quite right that we have proper legal processes, although sometimes one can get frustrated with how long they take."
Mrs May said the Government "will work to ensure that the suspects are handed over to the US authorities as quickly as possible". However the human rights judges warned the Government not to extradite any of the men until a three-month deadline for a final appeal has expired.
The verdict concerned five men including Hamza, currently serving a seven-year sentence in Britain for soliciting to murder and inciting racial hatred, and Babar Ahmad, a 36-year-old computer expert and alleged terrorism fundraiser who has been held in a UK prison without trial for nearly eight years.
Three others - Seyla Talha Ahsan, Adel Abdul Bary and Khaled Al-Fawwaz - can also be extradited, while the case of a sixth man, Haroon Rashid Aswat, was adjourned until a further hearing to consider his mental health state.
Hamza faces charges in the US on 11 counts of criminal conduct related to the taking of 16 hostages in Yemen in 1998, advocating violent jihad in Afghanistan in 2001 and conspiring to establish a jihad training camp in Bly, Oregon, between June 2000 and December 2001.
Aswat was indicted as Hamza's "co-conspirator", while Bary and Al-Fawwaz were indicted - along with Osama bin Laden and 20 others - for their alleged involvement in, or support for, the bombing of US embassies in Nairobi and Dar es Salaam in 1998. Al-Fawwaz faces more than 269 counts of murder. Ahmad and Ahsan are accused of offences including providing support to terrorists and conspiracy to kill, kidnap, maim or injure persons or damage property in a foreign country.
The judges acknowledged that, in America, Bary faced 269 mandatory sentences of life imprisonment without the possibility of parole, while Ahmad, Ahsan, Hamza and Al-Fawwaz faced "discretionary" life sentences.