This paper investigates the consistency of the least squares estimators and derives their limiting distributions in an AR(1) model with a single structural break of unknown timing. Let β1 and β2 be the preshift and postshift AR parameter, respectively. Three cases are considered: (i) |β1| < 1 and |β2| < 1; (ii) |β1| < 1 and β2 = 1; and (iii) β1 = 1 and |β2| < 1. Cases (ii) and (iii) are of particular interest but are rarely discussed in the literature. Surprising results are that, in both cases, regardless of the location of the change-point estimate, the unit root can always be consistently estimated and the residual sum of squares divided by the sample size converges to a discontinuous function of the change point. In case (iii), [circumflex over beta]2 does not converge to β2 whenever the change-point estimate is lower than the true change point. Further, the limiting distribution of the break-point estimator for shrinking break is asymmetric for case (ii), whereas those for cases (i) and (iii) are symmetric. The appropriate shrinking rate is found to be different in all cases.