The European Court of Human Rights has backed terror extraditions from the UK to the United States.
The Strasbourg court held there would be no violation of human rights for those facing life and solitary confinement in a supermax prison.
The court sanctioned the extradition of Abu Hamza, Babar Ahmad and three others.
Judges said they would consider further the case of another suspect because of mental health issues.
The court's ruling can still hypothetically be appealed to its final Grand Chamber - but in practice very few cases are reheard in that final forum.
The court's decision is one of its most important since 9/11 because it defines how and when the US can successfully seek the extradition of terror suspects from the UK.
The men have three months to try to persuade the Grand Chamber to reopen the entire case and examine it. If the men fail to launch an appeal, they will be extradited to the United States.
A spokesman for Babar Ahmad, who has been held for a record of nearly eight years without trial, said he would fight on against extradition.
Last week, he appealed in a BBC interview to be charged and tried in the UK because his alleged crimes were committed here.
The European Court said there would be no breach of human rights if the men were to be held at ADX Florence, a Federal Supermax jail used for people convicted of terrorism offences.
Abu Hamza is unlikely to be held at that jail because of his disabilities. The court also held that the life sentences each man faces would not breach human rights.
