We study a discontinuous Galerkin method for the Stokes equations with a new stabilization of the viscous term. On the one hand, it allows us to recover, as the stabilization parameter tends towards infinity, some stable and well-known nonconforming approximations of the Stokes problem. On the other hand, we can easily define an a posteriori error indicator, based on the reconstruction of a locally conservative H(div)-tensor. An a priori error analysis is also carried out, yielding optimal convergence rates. Numerical tests illustrating the accuracy and the robustness of the scheme are presented.