Claire Ovey, spokeswoman for the European Court of Human Rights, said Abu Hamza and the four other men set to be extradited to America have three months to appeal today's decision.
It will mean that Abu Hamza can now be extradited to the US, as his prison sentence in Britain comes to an end, to stand trial for alleged serious offences including supporting terror training camps in the United States.
Proper legal processes are important, but it remains a serious concern that this has taken so long, and it is important that remaining steps should be swiftly resolved.
The Home Secretary Theresa May has welcomed the decision by the European Court of Human Rights to allow the extradition of Abu Hamza and four other terror suspects. However, she told ITV News that it could take several months:
I welcome the long awaited decision by the European Court of Human Rights today. However, this basket of cases only goes to highlight the disarray of extradition and removals in the UK.
Babar Ahmad alone has waited eight years for a decision on his case. This delay is unacceptable.
We need a system where the Home Secretary is given more discretion over individual cases, ensuring that the important security cases are fast tracked. I hope she will introduce this when she reviews legislation in the summer.
Lord Carlile, the former independent reviewer of terrorism legislation, said the ruling by the European Court of Human Rights to allow the extradition of five British men to the US is "entirely sensible" but added it was "a shame" the decision had taken "so long" to happen.
The European Court of Human Rights' judges emphasised that today's ruling would only be final after three months - if there is no appeal - and the UK government cannot extradite the five men until that time.
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).
Babar is a British citizen accused of a crime said to have been committed in the UK and all the evidence against him was gathered in this country.
Nevertheless, British justice appears to have been subcontracted to the US.
This should be immediately rectified by putting Babar on trial in the UK and ordering a full public inquiry into the matter."
The decision to allow the extradition of five British nationals to America was a unanimous one from the European Court of Human Rights.
They said there would be no violation of Article 3 of the Human Rights Code - the "prohibition of inhuman and degrading treatment" - as a result of detention conditions the five men might face at ADX Florence "supermax" prison in the US.
The length of their possible sentences would not breach their human rights under European law either.
The Home Secretary Theresa May has said the government "will work to ensure that the suspects are handed over to the US authorities as quickly as possible".
I welcome the decision of the European Court of Human Rights to allow the extradition of Abu Hamza and other terror suspects.
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.
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.