The paper introduces an alternative estimator for the linear censored quantile regression model. The objective function is globally convex and the estimator is a solution to a linear programming problem. Hence, a global minimizer is obtained in a finite number of simplex iterations. The suggested estimator also applies to the case where the censoring point is an unknown function of a set of regressors. It is shown that, under fairly weak conditions, the estimator has a $\sqrt${n}-convergence rate and is asymptotically normal. In the case of a fixed censoring point, its asymptotic property is nearly equivalent to that of the estimator suggested by Powell (1984, 1986a). A Monte Carlo study performed shows that the suggested estimator has very desirable small sample properties. It precisely corrects for the bias induced by censoring, even when there is a large amount of censoring, and for relatively small sample sizes.