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The authors consider testing for a unit root in a time series with a structural change in mean occurring at an unknown date. They derive and tabulate the asymptotic distributions of the minimal t-statistic over all possible break points (and other related statistics) in the appropriate regressions. However, emphasis is given to the tabulation of finite sample critical values with particular attention given to the effect of various procedures to select the order of the estimated autoregressions. They apply the test to analyze the issue of purchasing power parity between the U.S. and the U.K. and Finland whose real exchange rates are characterized by apparent shifts in level when using particular price indices.