We consider a Cox model for right-censored survival data when a covariate is missing completely at random and when a continuous surrogate covariate and a complete validation sample are available. The likelihood is approximated using a nonparametric estimation of the missing covariate values and we propose new estimators of the regression parameter and of the cumulative hazard function based on the approximated likelihood. We show that the estimators are n 1/2 -consistent and asymptotically Gaussian and give conditions which ensure that the estimator of the regression parameter has the minimal asymptotic variance in the model with missing covariate data.