The Egyptian-born Abu Hamza is accused by Washington of supporting al Qaeda, aiding a kidnapping in Yemen and plotting to open a training camp for militants in the United States.
The 54-year-old also achieved notoriety for praising the Sept. 11, 2001 attacks on the United States and running a mosque in north London which Britain says acted as a hotbed of Islamist radicalism.
On Friday, two judges at the High Court dismissed his request to delay the proceedings in order to allow the cleric, who did not attend the hearing and is being held in a high-security British prison, to undergo a brain scan that his lawyers said would prove he was unfit to be extradited.
"Extradition may proceed immediately," they said in their ruling.
"It is unacceptable that extradition proceedings should take more than a relatively short time, to be measured in months not years," they added.
Four other terrorism suspects also have lost their appeals against extradition to the United States.
All five cases were sent to the High Court after the European Court of Human Rights refused to stop London extraditing them.
Britain's interior ministry would not say when Abu Hamza would be flown to the United States but it was expected to happen very soon. The BBC reported that two U.S. civilian jets, one of them registered to the U.S. Department of Justice, were on the tarmac at an air base in eastern England.
"We welcome the Court's decision and are working to extradite these men as quickly as possible," the Home Office said in a brief statement.
The United States seeks Abu Hamza as well as Khalid al-Fawwaz, Adel Abdul Bary, Babar Ahmad and Syed Talha Ahsan to face prosecution in U.S. courts on terrorism charges.
"These individuals are being transferred to the United States," the U.S. embassy said in a statement after the ruling. "These extraditions mark the end of a lengthy process of litigation through the UK courts and the ECHR."
As the judges read out their ruling, a crowd of about 100 people protested outside the London court, shouting "Free Abu Hamza" and holding banners reading "Stop extraditions" and "democracy = hypocrisy."
LONG BATTLE
A fiery anti-Western speaker, Abu Hamza is said to have inspired some of the world's most high-profile militants including Zacarias Moussaoui, one of the Sept. 11 conspirators.
He could now be sent to the ultra-secure "Supermax" prison in the United States with a sentence of more than 100 years.
The cleric was once a preacher at the Finsbury Park Mosque in north London but was later jailed in Britain for inciting murder and racial hatred.
Born Mustafa Kamel Mustafa, he moved to Britain as an engineering student in the 1970s, married a British woman and once worked as a doorman at discos in London.
Abu Hamza, who is famous for wearing a hook in place of his missing right hand, says he lost both hands and an eye while living in Afghanistan in the 1980s while carrying out humanitarian work. The authorities say he was fighting for the Mujahideen against the Soviet Union.
After being held on the U.S. extradition warrant, he was jailed for seven years by a British court in 2006 for inciting Muslims to kill Jews and non-believers, based on extracts of speeches he had given years earlier.
Hamza was indicted by a federal grand jury in new York in April 2004. He was accused of involvement in a 1998 hostage taking in Yemen which resulted in the deaths of four hostages - three Britons and one Australian.
He was also accused of providing material support to al Qaeda by trying to set up a training camp for fighters in Oregon in the United States and of trying to organise support for the Taliban in Afghanistan.
