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.
– Ashfaq Ahmad, father of Babar
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.
– David Cameron
I think deportation and extradition arrangements work promptly and properly particularly when people are accused of very serious crimes.
Former Liberal Democrat leader Sir Menzies Campbell gave a cautious welcome to the European Court of Human Rights' decision to allow the extradition of five British nationals to the US.
This decision will do a great deal to restore the reputation of the court but we must be cautious in drawing unjustified conclusions.
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."
The father of Babar Ahmad, Ashfaq, said he was "very disappointed" by the European Court of Human Rights' decision to allow his son to be extradited to the US.
Ashfaq Ahmad added that the situation needed to be "immediately rectified by putting Babar on trial in the UK."
The BBC's Home Affairs correspondent Dominic Casciani said the family of Babar Ahmed - one of the five men facing extradition to the US - will fight the European Court of Human Rights' decision.
The shadow immigration minister Chris Bryant has backed the decision by the European Court of Human Rights to allow the extradition of Abu Hamza and four others to America.
So the European Court backs extradition of Abu Hamza et al. Quite right too.From @ChrisBryantMP on Twitter: