The official statement said the convicted man had failed to meet a 30-day deadline to appeal against the sentence.
This is believed to be the first such punishment ordered since the introduction the Sharia penal code in the state earlier this year.
The state government said Buba Bello Kare Garke Jangebe - a Muslim man described as a notorious cow thief - had his right hand chopped off from the wrist on Wednesday.
Witnesses said a jubilant crowd watched the amputation carried out in the grounds of a state hospital in the town of Talata-Mafara.
"The people were happy that the norms of sharia were being carried out," a state government official was quoted as saying.
The official statement said the convicted man had failed to meet a 30-day deadline to appeal against the sentence.
Zamfara was the first state in Nigeria's Muslim-dominated north to introduce the Islamic penal code. Other nearby states have said they intend to follow.
Divisive
The moves towards Sharia have proved deeply divisive, being largely welcomed by Muslims but enraging Christians.
Last month, hundreds of people were killed in the northern city of Kaduna when violence began during an anti-Sharia demonstration by Christians.
The human rights organisation Huri Laws, has started court proceedings against the implementation of Sharia.
The organisation says Zamfara has violated the essentially secular nature of the Nigerian constitution through its adoption of Sharia, and that it is favouring one religion, Islam, to the detriment of the other, Christianity.
The Nigerian constitution specifically recognises the authority of Sharia on some aspects of family and civil law, but what has aroused such anger among Christians was Zamfara's move in January to introduce a Sharia criminal code.
Many Christians have argued that it is the government, through the attorney-general, which ought to be taking the pro-Sharia states to court.
But President Olusegun Obasanjo, who is a devout Christian, has said that any legal move by the government would be divisive.
He says he would prefer the government to remain neutral on this most delicate of subjects, while religious and secular leaders work on a political solution.