This article develops a semiparametric procedure to estimate parameters of an accelerated failure time model. To express the density of the error distribution, we use the P-spline (B-splines with penalties) smoothing technique. To accommodate error densities with infinite support (and for other reasons) we replace the B-splines with their limits as the degree of the B-spline goes to infinity; namely, with normal densities. The spline coefficients as well as any number of regression parameters are quickly and accurately estimated via penalized maximum likelihood. The method directly provides predictive survival distributions for fixed values of covariates while allowing for left-, right-, and interval-censored data. The approach has been implemented as an R package and is applied here to the problem of predicting AIDS-free survival in the presence of interval censoring.