In this case, in New York, the donor was tested 10 weeks before the surgery and the results were negative. But after the testing and before the surgery, he had unprotected sex with a man.
Neither the donor nor the recipient knew they had HIV until a year after the transplant.
The recipient developed AIDS, perhaps because he or she was on drugs that suppress the immune system to prevent organ rejection, while the donor did not, health officials said. Both are receiving HIV treatment.
This is the first documented case of its kind in the U.S. since screening for HIV began in the mid-1980s.
The Center for Disease Control and Prevention says better surveillance is needed. In a report Thursday, the CDC recommended that organ donors have repeat HIV tests a week before surgery.
"The most sensitive test needs to be done as close as possible to the time of transplant," said Dr. Colin Shepard, who oversees tracking of HIV cases for the New York City Health Department.
The CDC also said would-be organ donors should be told to avoid behavior that can increase their chances of infection.
Living organ donors in the U.S. are routinely tested for infectious diseases such as hepatitis and HIV. But the organization that oversees organ transplants in the U.S. does not have an explicit policy on when such screening should be done. That's left up to transplant centers.
The Associated Press contributed to this story