Two British Muslims should be extradited to the United States to stand trial on terrorist charges, the High Court ruled this morning.

The judgment is the most significant ruling yet in a terror case since the implementation of a new Anglo-American extradition treaty which speeded up the process of surrendering suspects.

Babar Ahmad and Haroon Rashid Aswat had argued that they ran the risk of being ill-treated or sent to the US internment camp at Guantanamo Bay if they were extradited.

But Lord Justice Laws, sitting with Mr Justice Walker, said the allegation that the US might violate undertakings about their treatment given to the British government "would require proof of a quality entirely lacking here".

The ruling, which could still be challenged in the House of Lords, also has implications for the case of Abu Hamza, the former imam of Finsbury Park mosque, who lost his appeal against conviction this week and is also facing extradition to the US.

Ahmad, 31, a computer expert from Tooting, south London, is accused of running jihadi websites which raised funds for terrorist organisations and incited people to wage holy war.

He is a cousin of Mohammed Naeem Noor Khan a Pakistani al-Qaeda operative whose arrest in 2004 led to the discovery of a plan to carry out car bomb attacks and detonate a "dirty bomb" in London.

Dhiren Barot, a British al-Qaeda member, was jailed for life for organising that conspiracy last month.

Aswat, 32, from Dewsbury, west Yorkshire, was radicalised by Abu Hamza and was one of the extremist clerics closest aides for a number of years.

He has been charged in the US on the same indictment as Abu Hamza over his alleged role in trying to set up a jihad training camp on a remote ranch in Bly, Oregon. In 1999 Abu Hamza sent Aswat and another man to America to visit the ranch and report back to him on its suitability.

The two men are alleged to have carried out firearms training and preached radical doctrines to a group of American Muslims.

Aswat was arrested in July 2005 in Zambia following a worldwide search for him in the aftermath of the 7/7 suicide bombings in London. But he was not charged with any offences here and the US sought his extradition.

The judges said they would take time to consider whether both men should be given permission to take their case to the House of Lords for a final ruling.

Edward Fitzgerald, QC, representing both men, argued that the case raised human rights issues of public importance which should go before the Law Lords.

Ashfaq Ahmad, the father of Babar Ahmad, said: "We are very disappointed with the High Court verdict today."

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