Observational studies are frequently conducted to compare the effects of two treatments on survival. For such studies we must be concerned about confounding; that is, there are covariates that affect both the treatment assignment and the survival distribution. With confounding the usual treatment-specific Kaplan-Meier estimator might be a biased estimator of the underlying treatment-specific survival distribution. This article has two aims. In the first aim we use semiparametric theory to derive a doubly robust estimator of the treatment-specific survival distribution in cases where it is believed that all the potential confounders are captured. In cases where not all potential confounders have been captured one may conduct a substudy using a stratified sampling scheme to capture additional covariates that may account for confounding. The second aim is to derive a doubly-robust estimator for the treatment-specific survival distributions and its variance estimator with such a stratified sampling scheme. Simulation studies are conducted to show consistency and double robustness. These estimators are then applied to the data from the ASCERT study that motivated this research.