In this paper, a class of cell centered finite volume schemes, on general unstructured meshes, for a linear convection-diffusion problem, is studied. The convection and the diffusion are respectively approximated by means of an upwind scheme and the so called diamond cell method [4]. Our main result is an error estimate oforder h, assuming only the W2,p (for p>2) regularity of thecontinuous solution, on a mesh of quadrangles. The proof is based on anextension of the ideas developed in [12]. Some newdifficulties arise here, due to the weak regularity of the solution, and thenecessity to approximate the entire gradient, and not only its normalcomponent, as in [12].