As her father lay dying at a hospital, Ms. Stepien said, her family “decided to accept the inevitable” and donated his organs through an organization called the Center for Organ Recovery and Education.

The organization allows donor families and the recipients to keep in touch with one another after the transplant. Mr. Stepien’s heart went to Arthur Thomas, a father of four who lives in Lawrenceville, N.J., and who Ms. Stepien said had been within days of dying.

Given a diagnosis of ventricular tachycardia about 16 years before receiving the transplant, Mr. Thomas, 72, said in an interview on Monday that he was in congestive heart failure when word arrived that his doctors had found a heart.

“In order to get to the top of the transplant list, you have to be really hurting,” Mr. Thomas said. “Once I had my transplant, I, of course, decided I would write a thank-you to the family.”

From there, a relationship was forged through monthly phone calls, emails and letters. Ms. Stepien’s mother, Bernice, kept in touch with Mr. Thomas, even swapping cards on Christmas and flowers on birthdays. At times, they compared parenting tips. But the families had not thought about meeting in person until Jeni Stepien, 33, became engaged to Paul Maenner, a 34-year-old engineer, in October.