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About 1.6 kilometers, or roughly one mile. See note at bottom.
Check out the links for more info on the Bering Strait, the narrowest part of which, East Cape (Deshnev) on the Siberian and Kamchatkan coast, approaches within about 90 kilometers (56 m.) of Cape Prince of Wales on the American Alaskan shore. The strait contains two small islands known as the Diomede Islands. These granite domes lie about 1.6 K, or one mile, apart and the boundary line between the possessions of Russia and the United States passes between them.
If you want to split hairs, then it's possible to be aboard a ship travelling between the two islands and at the moment the ship straddles the boundry the shortest distance between the two countries would be "one step".
Various reference sources use 2.5 and 3 miles as the shortest distances between the American and Russian islands.
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