"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.
Amnesty International condemned the sentence, handed down by a judge in Khartoum, as "appalling and abhorrent".
Local media report the sentence would not be carried out for two years after she has given birth.
Sudan has a majority Muslim population, which is governed by Islamic law. It rules that apostasy - the abandonment of one's religious faith - is a crime.
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.
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.
There were small groups of protesters outside the court - both her supporters and those who back the punishment.