Radical preacher Abu Hamza can be extradited to the United States to face terror charges, a European court has ruled.
The unanimous ruling by the European Court of Human Rights said five British suspects, including Hamza, would not be subject to "ill-treatment" in America.
Hamza is wanted in America on 11 charges related to taking 16 hostages in Yemen in 1998, promoting violent jihad in Afghanistan in 2001 and conspiring to set up a jihad training camp in Oregon, America.
Also among those facing extradition alongside him is Babar Ahmad, a 36-year-old computer expert who has been in a UK prison without trial for nearly eight years. He has been refused bail since his arrest in August 2004 on a US extradition warrant.
MORE DETAILS: Read Longer Extracts From Today's Judgement
The three other British nationals who lost their appeals were Seyla Talha Ahsan, Adel Abdul Bary and Khaled Al-Fawwaz.
The five men could now appeal to the Grand Chamber at The European Court of Human Rights, it is believed - but few cases are ever heard in that forum, the BBC said.
The European court adjourned its decision on another suspect, Haroon Rashid Aswat, pending consideration of further complaints lodged by him.
Aswat's case was adjourned "as it required further submissions... on the relevance of his schizophrenia and detention at Broadmoor Hospital to his complaint concerning detention at ADX".
Speaking in Japan, Prime Minister David Cameron said he was "very pleased" by the decision made in Strasbourg:
"I'm very pleased with the news. It is quite right to have a proper legal process although sometimes one can get frustrated with how long they take.
"I think deportation and extradition arrangements work promptly and properly particularly when people are accused of very serious crimes."
Home Secretary Theresa May said she welcomed the decision to allow the extradition and said the suspects would be sent to the US "as quickly as possible".
In five of the six cases, the court found that extradition would not breach their human rights and, in the remaining case, it asked for further information before taking a final decision.
Tuesday's verdict declared that "detention conditions and length of sentences of five alleged terrorists would not amount to ill-treatment if they were extradited to the USA".
The human rights judges emphasised that today's ruling only becomes final after three months, if there has been no further appeal.
Meanwhile, the judges said, "the court decided to continue its indication to the United Kingdom Government that the applicants should not be extradited until this judgment became final or until the case was referred to the Grand Chamber (of the Human Rights court)".
The family of Babar Ahmad said they were "very disappointed" with the ruling.
Speaking at their home Babar's father Ashfaq said that his son would continue to fight to stand trial in the UK, and that they understood he had three months to appeal.
He said that the situation needed to be "immediately rectified by putting Babar on trial in the UK".
"The fundamental question remains as to why this matter has even got to Strasbourg, and why Babar needs to be extradited to the US," he said. "There has been a serious abuse of process."
The ruling was given a cautious welcome by former Liberal Democrat leader Sir Menzies Campbell, who said it would be important to study the detailed judgment.
"This decision will do a great deal to restore the reputation of the court but we must be cautious in drawing unjustified conclusions," he said.
"The detail and reasoning of the judgment has to be carefully examined but perhaps now we can have a rational debate about the role and significance of the European Convention and its fundamental importance to a democratic society like our own."
All six suspects have been indicted on various charges of alleged terrorism in America and argue that, if extradited and convicted in America, their conditions of detention if held at one of the country's so-called "supermax" prisons would amount to ill-treatment under Article 3 of the Human Rights code.
The European Court of Human Rights halted extradition proceedings in July 2010, arguing it needed more time to consider whether the moved risked breaching the suspects' rights by exposing them to possible life imprisonment without parole and solitary confinement.
The 53-year-old Hamza, born in Alexandria, studied civil engineering and in 1984 he married a British woman, Valerie Fleming.
But throughout the 1980s, he slowly began to turn towards a fundamentalist interpretation of the Koran.
In February 2006, he was jailed in the UK for seven years for soliciting murder and inciting racial hatred.
He is being held in the maximum-security Belmarsh prison in Woolwich, south-east London.
The American authorities have described him as a "terrorist facilitator with a global reach". But before the US extradition request could be dealt with, Abu Hamza had been convicted in 2006 of terrorist-related charges in Britain.
Human Rights Extradition Row
The case of Hamza, described by the American authorities as a "terrorist facilitator with a global reach", was set to reignite the row over the role of the human rights court if today's verdict blocked extradition.
After a series of Strasbourg human rights rulings against the Government, ministers were certainly braced for another rebuff.
The fact that the judges backed the UK - and America - still leaves unfinished business, including an instruction from the human rights court to restore the right to vote to at least some prisoners in the UK.
And a controversial verdict in January blocking the deportation to Jordan of terror suspect Abu Qatada is still being challenged.
Last month Home Secretary Mrs May went to Jordan, seeking the formal assurances the human rights court is demanding that Qatada would not be tried using evidence which could have been obtained by torture.
She returned insisting that talks with the Jordanians had been "positive", but with nothing yet to offer the Strasbourg court.
The judgment went on: "In particular, not all inmates convicted of international terrorism were housed at ADX and, even if they were, sufficient procedural safeguards were in place, such as holding a hearing before deciding on such a transfer.
"Furthermore, if the transfer process had been unsatisfactory, there was the possibility of bringing a claim to both the Federal Bureau of Prisons' administrative remedy programme and the US federal courts."
On the jail's "restrictive conditions and lack of human contact", the judges said: "The court found that, if the applicants were convicted as charged, the US authorities would be justified in considering them a significant security risk and in imposing strict limitations on their ability to communicate with the outside world.
"Besides, ADX inmates - although confined to their cells for the vast majority of the time - were provided with services and activities (television, radio, newspapers, books, hobby and craft items, telephone calls, social visits, correspondence with families, group prayer) which went beyond what was provided in most prisons in Europe."
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.
The judges declared: "Having regard to the seriousness of the offences in question, the court did not consider that these sentences were grossly disproportionate or amounted to inhuman or degrading treatment. There would therefore be no violation of Article 3 in the case of any of these five applicants if they were extradited, convicted and given life sentences."