Justyna and Rafal Sliwa told the New York Post they were notified of the mix-up July 6.
They held a memorial service in Brooklyn on June 16 for a stillborn they believed to be theirs. The funeral home sent the remains to the couple's hometown of Lomza, Poland. A funeral was held there June 19.
Information on the other family was not made public. A spokeswoman for the medical examiner's office, Ellen Borakove, says the babies had "strikingly similar names."
The office is investigating how the mix-up occurred.
Borakove calls it a "tragic, unfortunate error we have to make right."