The family of Anni Dewani said they have waited "three years, three months" for a ruling that paves the way for her husband Shrien Dewani to be extradited to South Africa to stand trial for her 2010 murder.
Speaking on behalf of family members at London's High Court, Mrs Dewani's first cousin Amit Karia said: "We are happy with the judgment. We have waited for it for three years, three months. .... It seems a step closer to finally getting justice for our Anni."
But he warned there was still a chance that Shrien Dewani could launch a last-ditch appeal over today's ruling to the Supreme Court, the highest court in the land.
Mr Karia said: "We are always worried and don't take anything for granted."
He also described the situation for Anni's parents as "immensely difficult - they do not have normal life any more," adding: "They just want the truth - that is all we have ever wanted."
Today's High Court rejection of Shrien Dewani's bid to delay his extradition to South Africa is the latest ruling in a lengthy legal process.
Dewani's extradition to eventually stand trial for his wife's killing in November 2010 has been delayed while he receives treatment for mental health problems.
The go-ahead for Dewani's extradition was originally granted in 2011 before the High Court allowed an appeal.
Last July, chief magistrate Howard Riddle ruled at Westminster Magistrates' Court that Dewani should be extradited and rejected his attempt to stay in the UK for further hospital treatment.
But Dewani then won a further High Court hearing, centred on two legal issues.
The first related to Dewani's status as "an accused person" while the second concerned whether it would be "unjust and oppressive" to extradite him "regardless of the prognosis" of his mental condition. The High Court has today ruled against him on those matters.
Three men have already been convicted over Anni Dewani's death.
South African Xolile Mngeni was convicted of premeditated murder for shooting her. Prosecutors claimed he was a hitman hired by Shrien Dewani to kill his wife, which Dewani has consistently denied.
Taxi driver Zola Tongo was jailed for 18 years after he admitted his part in the killing and another accomplice, Mziwamadoda Qwabe, also pleaded guilty to murder and was handed a 25-year prison sentence.
A panel of three judges has ruled it would not be "unjust and oppressive" to extradite Honeymoon murder suspect Shrien Dewani if the South African government could confirm how long he would be kept in the country without standing trial.
The High Court heard today that the government indicated it was willing to give that undertaking.
Dewani, who is compulsorily detained in hospital under the Mental Health Act, is accused of ordering the killing of his new wife Anni, 28, who was shot as the couple travelled in a taxi on the outskirts of Cape Town in November 2010.
Dewani, from Bristol, has been fighting removal from the UK to face proceedings over his wife's death until he has recovered from mental health problems, including depression and post-traumatic stress disorder.