Amnesty International condemned the sentence, handed down by a judge in Khartoum, as "appalling and abhorrent".
Local media report the sentence on the woman, who is pregnant, would not be carried out for two years after she had given birth.
Sudan has a majority Muslim population, which is governed by Islamic law.
"We gave you three days to recant but you insist on not returning to Islam. I sentence you to be hanged to death," the judge told the woman, AFP reports.
Western embassies and rights groups had urged Sudan to respect the right of the pregnant woman to choose her religion.
The judge also sentenced the woman to 100 lashes after convicting her of adultery - because her marriage to a Christian man was not valid under Islamic law.
This will reportedly be carried out when she has recovered from giving birth.
Earlier in the hearing, an Islamic cleric spoke with her in a caged dock for about 30 minutes, AFP reports.
Then she calmly told the judge: "I am a Christian and I never committed apostasy."
Amnesty International said the woman, Meriam Yehya Ibrahim Ishag, was raised as an Orthodox Christian, her mother's religion, because her father, a Muslim, was reportedly absent during her childhood.
In court, the judge addressed her by her Muslim name, Adraf Al-Hadi Mohammed Abdullah.
A Sudanese court has sentenced a woman to hang for apostasy - the abandonment of her religious faith - after she married a Christian man.