Nevertheless, the influx is testing the limits of tolerance and hospitality in Sumte, and across Germany. It is also straining German politics broadly, creating deep divisions in the conservative camp of Chancellor Angela Merkel and energizing a constellation of extremist groups that feel their time has come.

One of the few people, in fact, who seem enthusiastic about the plan for Sumte is Holger Niemann, 32, an admirer of Hitler and the lone neo-Nazi on the elected district council. He rejoices at the opportunities the migrant crisis has offered.

“It is bad for the people, but politically it is good for me,” Mr. Niemann said of the plan, which would leave the German villagers outnumbered by migrants by more than seven to one.

Germans face “the destruction of our genetic heritage” and risk becoming “a gray mishmash,” Mr. Niemann added, predicting that public anxiety over Ms. Merkel’s open-armed welcome to refugees would help demolish a postwar political consensus in Germany built on moderation and compromise.