HANDEN, Sweden—Hundreds of young unmarried Mormons descended on a high-school gymnasium here on Stockholm's outskirts late last month for a raucous dance party that would continue until sunrise.

No alcohol was served, but one would hardly have known. By midnight, the dancing reached a fever pitch. A pack of rowdy Britons in T-shirts emblazoned with “I [heart] Swedish Girls” roamed the floor. Outside, a group attempted something akin to a keg stand with a water cooler.

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