We consider a time-dependent and a steady linear convection-diffusion equation. These equations are approximately solved by a combined finite element -- finite volume method: the diffusion term is discretized by Crouzeix-Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the unsteady case, the implicit Euler method is used as time discretization. This scheme is shown to be unconditionally L2-stable, uniformly with respect to the diffusion coefficient.