Abu Hamza was well known as a radical preacher before his arrest
Radical Muslim cleric Abu Hamza al-Masri has been refused leave to appeal to the House of Lords against his extradition to the US.
US prosecutors want the 50-year-old to face trial on charges which include providing support to al-Qaeda.
The Egypt-born preacher is currently serving a seven-year jail term in the UK for inciting murder and race hate.
BBC correspondent Andy Tighe said his lawyers are now likely to take the case to the European Court of Human Rights.
The Court of Appeal ruling comes after Home Secretary Jacqui Smith formally approved the extradition in February.
Lawyers for Abu Hamza, from west London, have been fighting his extradition since the request was first filed in May 2004.
US prosecutors have outlined 11 charges, alleging he funded terrorism, organised a training camp in Oregon between 1998 and 2000, and conspired to take 12 Westerners hostage in Yemen in 1998.
The extradition process was put on hold after Abu Hamza was arrested by Metropolitan Police officers in 2004.
In February 2006, an Old Bailey jury found him guilty of charges related to speeches at Finsbury Park Mosque in north London.
Extradition proceedings began again in May 2007 and a district judge later approved the request.
The High Court turned down Abu Hamza's appeal against the decision earlier this year, rejecting the argument that the US evidence had been obtained by torture, and was therefore inadmissible.
At the latest Appeal Court hearing, senior judge Sir Igor Judge refused to certify that the case raised a point of law of public importance to go before the Law Lords.
"The 2003 Extradition Act must be applied and our extradition commitments honoured," he said.