A man who agreed to donate part of his liver to help a sick relative died while undergoing the transplant procedure at Lahey Clinic in Burlington two weeks ago, the hospital said yesterday.

It was only the third death of an adult living liver donor in the United States in the two decades since the first procedure was done, according to two leading transplant surgeons. A total of 4,036 have been performed.

While any surgery carries risks, specialists said the death of a living donor is especially upsetting because they are generally young and healthy and are undergoing an operation they do not need for the benefit of a family member or close friend.

Lahey, which has transplanted more livers from living donors than any other US hospital, declined to identify the donor or release further details of the May 24 fatality, including the condition of the recipient.

“Families have informed the clinic of their desire to grieve privately,’’ Lahey said in a statement.

“Our deepest sympathies are with the families involved,’’ Dr. David Barrett, the clinic’s chief executive, said in an e-mail to employees. “Our thoughts today are also with the surgical team. They are extraordinary caregivers dedicated to healing patients even in the most trying circumstances. While we are saddened by this loss, we must continue our mission to offer life-saving treatment options like this to our patients.’’

Lahey is conducting a review of what went wrong, and the hospital reported the death to the Massachusetts Department of Public Health, which has opened an investigation. Department spokeswoman Julia Hurley said officials could not comment until the review is complete.

Lahey also reported the death to the United Network for Organ Sharing, which oversees organ transplants in the United States and is conducting its own review.

Liver transplants from living donors are commonly done in certain parts of the country, including New England, where the number of ill patients awaiting transplants greatly exceeds the supply of cadaver donors. Doctors usually remove about 60 percent of the donor’s liver and transplant it into the recipient. Within several weeks, the liver in both patients almost completely regenerates.

Hospitals screen donors extensively for several months for physical and psychological health and to make sure they understand the risks and are making the decision without pressure and in an ethical manner.