Lawyers for Dewani, who has post-traumatic stress disorder, argued that he should not be sent there until he is fit to plead over the murder of his Swedish wife Anni in a township outside Cape Town in November 2010.
But judges ruled it would not be "unjust and oppressive" to extradite him if the South African government agrees to return him to Britain if he never recovers sufficient health to go on trial., The court heard that South Africa has indicated it was willing to give such a guarantee and has 14 days to do so.
Dewani has been fighting extradition from Britain to face proceedings over his wife's death until he has recovered from mental health problems, including depression and post-traumatic stress disorder.
Dewani, who has been committed to a mental hospital, is accused of ordering the killing of his 28-year-old wife, who was shot as the taxi the couple were travelling in was apparently carjacked.
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Dewani denies any involvement in the killing.
A South African man, Xolile Mngeni, was jailed for life for the murder last December. Two other local men jailed over the killing allege that Dewani ordered the hit.