Standard randomized trials may have lower than desired power when the treatment effect is only strong in certain subpopulations. To address such situations, we develop a new trial design that combines two types of pre-planned rules for updating how the trial is conducted based on data accrued during the trial. The first component of our design involves response-adaptive randomization, in which the probabilities of being assigned to the treatment or control arm are updated during the trial to target an optimal allocation. The second component of our design involves two-stage, adaptive enrichment, where the enrollment criteria may be restricted to a subpopulation. We focus on the case of two subpopulations, e.g., defined by disease severity or a biomarker measured at baseline. The goals of the design include the following: to increase power, to increase the number of participants assigned to the superior treatment arm, and to control the familywise Type I error rate. We do a simulation study to compare our response-adaptive enrichment design to three simpler designs: a standard randomized trial design, a response-adaptive design, and an enrichment design. Our simulation study compares these designs in scenarios that arise from the problem of testing the effectiveness of a hypothetical new antidepressant.