At the corner where East North Street meets North Cherry Street in the small Ohio town of Kenton, the Immaculate Conception Church keeps a handwritten record of major ceremonies. Over the last decade, according to these sacramental registries, the church has held twice as many funerals as baptisms.

In tiny communities like Kenton, an unprecedented shift is under way. Federal and other data show that in 2013, in the majority of sparsely populated U.S. counties, more people died than were born—the first time that’s happened...