While undergoing surgery to donate a kidney to a male relative, the woman's aorta was cut or ruptured, causing her to bleed to death, the New York Post reported.
She was the mother of a 2-year-old child.
The program was more than 40-years-old. The state Health Department considers the death a "major incident" and has since initiated an investigation.
The Health Department did not order the program to be suspended. The hospital chose to do so while the investigation is under way.
"This is a terrible tragedy. The patient experienced a rare complication of this surgery," a hospital spokesman told The Post. "The doctors recognized the problem and took extensive steps to save the patient's life."