The case, thought to be the first of its kind to be heard in Sudan, involves a woman whose Christian name is Mariam Yahia Ibrahim Ishag.
“We gave you three days to recant but you insist on not returning to Islam,” Judge Abbas Mohammed Al-Khalifa told the woman on Thursday, addressing her by her father’s Muslim name, Adraf Al-Hadi Mohammed Abdullah.
“I sentence you to be hanged to death.”
Khalifa also sentenced Ishag to 100 lashes for “adultery”.
Ishag, who rights activists say is pregnant and 27 years old, reacted without emotion when Abbas delivered the verdict at a court in the Khartoum district of Haj Yousef.
Earlier in the hearing, an Islamic religious leader spoke with Ishag in the caged dock for about 30 minutes.
Then she calmly told the judge: “I am a Christian and I never committed apostasy.”
Sudan’s government introduced Islamic law in 1983 but extreme punishments other than flogging are rare.
After the verdict, about 50 people demonstrated against the decision.
“No to executing Mariam,” said one of their signs while another proclaimed: “Religious rights are a constitutional right.”
In a speech, one demonstrator said they would continue their activism with sit-ins and protests until she is freed.
“The details of this case expose the regime’s blatant interference in the personal life of Sudanese citizens,” Sudan Change Now Movement, a youth group, said in a statement on Wednesday.
A smaller group supporting the verdict also arrived but there was no violence.
“This is a decision of the law. Why are you gathered here?” one supporter asked, prompting an activist to retort: “Why do you want to execute Mariam? Why don’t you bring corruptors to the court?”
Speaking to AFP news agency on Wednesday, Ahmed Bilal Osman, Sudan’s information minister, said: “It’s not only Sudan. In Saudi Arabia, in all the Muslim countries, it is not allowed at all for a Muslim to change his religion.”