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In randomized controlled trials, treatment is often assigned by stratified randomization. I show that among all stratified randomization schemes that treat all units with probability one half, a certain matched-pair design achieves the maximum statistical precision for estimating the average treatment effect. In an important special case, the optimal design pairs units according to the baseline outcome. In a simulation study based on datasets from ten randomized controlled trials, this design lowers the standard error for the estimator of the average treatment effect by 10 percent on average, and by up to 34 percent, relative to the original designs.