CALEXICO, Calif.—Along a dusty stretch of border, flanked by an outlet mall and farmland, a giant crane hoisted a 30-foot-high section of steel barrier and dropped it into a shallow trench last Wednesday.

The structure was one of the first new pieces of a border wall put in place since the start of a lengthy construction project in February just west of the border crossing in downtown Calexico, a small, mostly Hispanic farming town.

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