Afghanistan has carried out its first execution since the fall of Taleban hardliners more than two years ago.
President Karzai authorised the execution
A former military commander convicted of murder was killed at a jail outside Kabul last week, it emerged on Tuesday.
Abdullah Shah received a single shot to the head after President Karzai gave his approval, the attorney general's office told the Associated Press.
Amnesty International, the human rights group, says Abdullah Shah was denied even basic standards of fairness.
The group said it feared the "execution may have been an attempt by powerful political players to eliminate a key witness to human rights abuses".
Abdullah Shah served under another commander, Zardad, in the 1992-96 civil war, and earned the nickname Zardad's Dog for attacks on travellers along the road between Jalalabad and Kabul in the 1990s.
The BBC's Andrew North in Kabul says the fact that the government has only just confirmed an execution which took place on 20 April shows just how sensitive it is to criticism about the case.
Afghan officials say Abdullah Shah, executed at Pul-e-Charkhi jail, was convicted on 20 counts of murder in special court proceedings in October 2002.
He was found guilty of killing one of his wives by pouring boiling water over her body.
Another wife, who said he had tried to burn her to death after dousing her with petrol, was one of those who testified against him.
Abdullah Shah was being held in the feared Pul-e-Charkhi prison
The court heard Abdullah Shah murdered his baby daughter by banging her repeatedly against a wall, officials say.
President Karzai had signed the death warrant reluctantly, his spokesman, Jawed Ludin, said.
But he added: "The president felt compelled by the need to ensure justice to the victims, especially in view of the nature of the crimes he [Abdullah Shah] committed."
The president had ordered a review of the case when the guilty verdict had first been delivered, but justice demanded the verdict "be delayed no further", the spokesman said.