A Bayesian bootstrap for a censored data model is introduced. Its small sample distributional properties are discussed and found to be similar to Efron's bootstrap for censored data. In the absence of censoring, the Bayesian bootstrap for censored data reduces to Rubin's Bayesian bootstrap for complete data. A first-order large-sample theory is developed. This theory shows that both censored data bootstraps are consistent bootstraps for approximating the sampling distribution of the Kaplan-Meier estimator. It also shows that both bootstraps are consistent bootstraps for approximating a posterior distribution of the survival function with respect to each member of the class of conjugate beta-neutral process priors.