Born to a Muslim father, the woman was convicted yesterday under the Islamic sharia law that has been in force in Sudan since 1983 and outlaws conversions on pain of death.
Meriam Yahia Ibrahim Ishag, 27, is married to a Christian and eight months pregnant, human rights activists say.
"We gave you three days to recant but you insist on not returning to Islam. I sentence you to be hanged," Judge Abbas Mohammed Al-Khalifa told the woman, addressing her by her father's Muslim name, Adraf Al-Hadi Mohammed Abdullah.
In Washington, the State Department said the United States was "deeply disturbed" by the sentence and urged Sudan to protect freedom of religion.
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Britain's Minister for Africa, Mark Simmonds, said he was "truly appalled".
"This barbaric sentence highlights the stark divide between the practices of the Sudanese courts and the country's international human rights obligations," he said in a statement.
Ishag, dressed in traditional Sudanese robes with her head covered, reacted without emotion when the verdict was read out at a court in the Khartoum district of Haj Yousef, where many Christians live.
But she calmly told the judge: "I am a Christian and I never committed apostasy."
Sudan has an Islamist government but, other than floggings, extreme sharia law punishments have been rare.
"The fact that a woman has been sentenced to death for her religious choice, and to flogging for being married to a man of an allegedly different religion, is appalling and abhorrent," said Amnesty International's Sudan researcher, Manar Idriss.
If the death sentence is carried out, she will be the first person executed for apostasy under the 1991 penal code, said Christian Solidarity Worldwide, a British-based campaign group.