MOSCOW—The atmosphere felt like a festival outside the Potomac auto-repair garage on Kiev Street in Moscow, where thousands of Ukrainians originally from the southeast Donetsk and Luhansk regions showed up to cast their ballots at a makeshift absentee polling station.

Voters lined up in a queue that stretched around the block. Many cheered, sang and held up handwritten signs bearing the names of their hometowns, as they waited to vote in a referendum that Ukrainian, European and American authorities denounced as a farce.

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