SUMTE, Germany — This bucolic, one-street settlement of handsome redbrick farmhouses may for the moment have many more cows than people, but next week it will become one of the fastest growing places in Europe. Not that anyone in Sumte is very excited about it.

In early October, the district government informed Sumte’s mayor, Christian Fabel, by email that his village of 102 people just over the border in what was once Communist East Germany would take in 1,000 asylum seekers.

His wife, the mayor said, assured him it must be a hoax. “It certainly can’t be true” that such a small, isolated place would be asked to accommodate nearly 10 times as many migrants as it had residents, she told him. “She thought it was a joke,” he said.

But it was not. Sumte has become a showcase of the extreme pressures bearing down on Germany as it scrambles to find shelter for what, by the end of the year, could be well over a million people seeking refuge from poverty or wars in Africa, Syria, Iraq, Afghanistan and elsewhere.