Radical Muslim cleric Abu Hamza loses his legal battle to remain in the UK and will be sent to the US.
A panel of European judges rejected his request for an appeal over a European court of human rights (ECHR) ruling that his extradition would not breach his rights.
Terror suspect Babar Ahmad will also be extradited.
The decision means Hamza, who was jailed for seven years for soliciting to murder and inciting racial hatred, will now be extradited with four others, including computer expert Ahmad, who has been held in a UK prison without trial for eight years after being accused of raising funds for terrorism.
The other cases involve Seyla Talha Ahsan, Adel Abdul Bary and Khaled Al-Fawwaz.
The ECHR said on 10 April 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 judges said that between 1999 and 2006 the men were indicted on various terrorism charges in America. They are likely to be detained at ADX Florence "supermax" prison in Colorado.
Hamza has been charged with 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.
Newcastle-based businessman Karl Watkin said earlier this month that he wanted to bring a private prosecution against Ahmad, pictured above, rather than see him extadited to the US.
Ahmad's family said his extradition should be halted until a decision is made on this private prosecution, adding in a statement: "The decision of the Grand Chamber is largely irrelevant to us as this matter should never have come to this stage had the British police done their job almost nine years ago and provided the material seized from Babar's home to the CPS (Crown Prosecution Service) rather than secretly passing it to their US counterparts.
"The CPS is now in possession of all that material which forms the basis of the US indictment and should immediately prosecute Babar for conduct allegedly committed in the UK.
"There is enormous public interest in Babar being prosecuted in the UK, as reflected by the fact that almost 150,000 members of the British public signed a government e-petition to this effect last year.
A Home Office spokesman said: "The home secretary welcomes today's decision not to refer the cases of Abu Hamza and four others to the grand chamber.
"This follows the judgment of the European court of human rights on April 10 to allow the extradition of these five terrorism suspects to the US. We will work to ensure that the individuals are handed over to the US authorities as quickly as possible."