The present work concerns the numerical approximation of the $M_1$ model for radiative transfer. The main purpose is to introduce an accurate finite volume method according to the nonlinear system of conservation laws that governs this model. We propose to derive an HLLC method which preserves the stationary contact waves. To supplement this essential property, the method is proved to be robust and to preserve the physical admissible states. Next, a relevant asymptotic preserving correction is proposed in order to obtain a method which is able to deal with all the physical regimes. The relevance of the numerical procedure is exhibited thanks to numerical simulations of physical interest.