This paper discusses shrinkage estimation in nonparametric Bayesian survival analysis using censored data. The shrinkage estimators proposed are based on estimating the parameter measure of a prior Dirichlet process in a nonparametric Bayesian survival curve estimator which is the posterior mean of this process. The shrinkage is toward a prior family of exponential survival curves. The estimators are then compared by simulation with the wholly nonparametric estimator of Kaplan-Meier and the maximum likelihood estimator for the exponential family. These comparisons are done in cases where the exponential assumption is both