The success of a new drug is assessed within a clinical trial using a primary endpoint, which is typically the true outcome of interest—for example, overall survival. However, regulators sometimes approve drugs using a surrogate outcome—an intermediate indicator that is faster or easier to measure than the true outcome of interest—for example, progression-free survival—as the primary endpoint when there is demonstrable medical need. Although using a surrogate outcome (instead of the true outcome) as the primary endpoint can substantially speed up clinical trials and lower costs, it can also result in poor drug-approval decisions because the surrogate is not a perfect predictor of the true outcome. In this paper, we propose combining data from both surrogate and true outcomes to improve decision making within a late-phase clinical trial. In contrast to broadly used clinical trial designs that rely on a single primary endpoint, we propose a Bayesian adaptive clinical trial design that simultaneously leverages both observed outcomes to inform trial decisions. We perform comparative statics on the relative benefit of our approach, illustrating the types of diseases and surrogates for which our proposed design is particularly advantageous. Finally, we illustrate our proposed design on metastatic breast cancer. We use a large-scale clinical trial database to construct a Bayesian prior and simulate our design on a subset of clinical trials. We estimate that our design would yield a 16% decrease in trial costs relative to existing clinical trial designs, while maintaining the same Type I/II error rates. This paper was accepted by J. George Shanthikumar for the Special Issue on Data-Driven Prescriptive Analytics.