A discretization of heterogeneous and anisotropic diffusion problems on general discretization meshes is developed and studied. The unknowns of this scheme are the values at the center of the control volumes and at some internal interfaces, chosen because of some irregularity of the diffusion tensor. If the tensor is regular enough, the values on the interfaces may be deduced from the values at the center, at the expense of loosing the local conservativity of the fluxes. This scheme is shown to be accurate on several numerical examples. Mathematical convergence to the continuous solution is obtained for homogeneous and heterogeneous tensors. An error estimate may be drawn under sufficient regularity assumptions on the solution.