Virtua Our Lady of Lourdes Hospital in Camden carried out a transplant intended for another patient with the same name and of a similar age.
The mistake, which the hospital management described as an "unprecedented event" was discovered a day after the transplant procedure.
"Mistakes of this magnitude are rare," Reginald Blaber, executive vice president and chief clinical officer at Virtua Health, said in the statement. "And despite the unusual circumstances of similar patient identities, additional verification would have prevented this error."
"As an organisation committed to safety and process, we immediately instituted additional measures and educational reinforcement to help ensure this does not happen again."
A member of the hospital's clinical team realised a day after the first transplant on 18 November that the 51-year-old received the kidney out of priority order, as they were lower on the transplant matching list, which is managed by the United Network for Organ Sharing.
The patient who was supposed to receive the initial kidney received another one about a week later and both patients are doing fine.
Virtua Our Lady of Lourdes Hospital is the only facility in southern New Jersey to provide kidney, liver and pancreas transplants.
About 113,000 people are on a waiting list for an organ transplant, with nearly 95,000 of them waiting for kidneys.
The hospital has not identified the patients.