Karl Hilsenbek is worried. “No one knows what will happen tomorrow, or where migrant numbers go from here,” says the mayor of Ellwangen, a flourishing town in Germany’s prosperous southwestern state of Baden-Württemberg. Unemployment there is under 2 percent. Large billboards advertise companies searching for new trainees. Little cause for worry, one would think.

But on the outskirts of town, on the grounds of an old army barracks, lies a …