In a blog posting, the children of Jerry and Barbara Heil of White Bear Lake, Minn., say they are "certainly disheartened to hear the news" but understand.
Italian emergency officials decided to end the search due to the danger to rescue workers. The Heils are the only Americans missing in the Jan. 13 wreck.
"We cannot express enough our sincere gratitude to all those involved in the search and rescue effort. Time and time again, the rescuers faced many perils in the hopes of reuniting the missing with their families. We will be forever grateful for all those who worked so hard for people they did not even know, yet understood how important their job was for those that remained waiting for news," the Heil family said in a statement.
"As we struggle to come to grips with this tragedy, we find comfort knowing Mom and Dad are now in a better place free from any worries. They have always been obedient to God's plan and now we must do the same."
Some 4,200 passengers and crew were on board the Costa Concordia cruise ship when it capsized. Seventeen bodies have been recovered, of which one has not yet been identified. Sixteen people are listed as missing but are presumed dead. The last time anyone was found alive was Jan. 15.
The Heils are active members of the Church of St. Pius X in White Bear Lake, a St. Paul suburb. Church members describe the Heils as quiet, kind people deeply involved in their congregation. Jerry Heil, 69, who retired from a job at the Minnesota Department of Agriculture, would teach religious education classes, and Barbara Heil, 70, would bring baked goods to class members.
Diane Vorwald, 48, who uses a wheelchair, said Jerry Heil would come to her home weekly to administer communion. In the summer, he would bring her fresh pies, she recalled.
"He's done a lot for me. It's a loss to me," Vorwald said.
The Heil family is making plans for a memorial service.
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