Cheney has had five heart attacks, the first of which was in 1978 when he was 37 years old, and the fourth in November 2000, after he and former President George W. Bush were elected to the White House. In 2001, Cheney had a pacemaker installed into his chest, and in September 2009, he underwent elective back surgery to treat lumbar spinal stenosis.
Walsh expects Cheney will have an easy recovery from the transplant partly because "he is used to heart surgeries."
Cheney's family said in a statement: "Although the former Vice President and his family do not know the identity of the donor, they will be forever grateful for this lifesaving gift.
"Former Vice President Cheney is thankful to the teams of doctors and other medical professionals at Inova Fairfax and George Washington University Hospital for their continued outstanding care."
Mitt Romney tweeted, "Ann and I send our thoughts and prayers to Vice President Cheney for a fast and full recovery."
Dr. Yana Svetlichnaya, an internal medicine resident at Northwestern University's McGaw Medical Center, contributed to this story.