The censored least absolute deviation estimator (CLAD) of censored regression models allows estimation under weak identification assumptions, but suffers from poor finite-sample performance, especially in very small samples. This gave rise to stepwise estimation methods of the median regression line under censoring that typically identify the subset of observations, where the conditional median is not censored, and employ the standard least absolute deviation (LAD) estimator for the selected subset afterwards. This paper introduces an alternative semiparametric estimator, which does not involve the stepwise selection procedure of a subset of data. Instead, the LAD estimator is applied directly to the complete (censored) data set and its bias is corrected so that the estimates become consistent and asymptotically normal under some regularity conditions. Besides deriving its asymptotic distribution, the finite-sample behavior is studied and compared to existing methods by means of Monte Carlo experiments.