A four-dimensional potential energy surface (PES) for the CO dimer consisting of rigid molecules has been calculated, using a scheme that combines density functional theory to describe the monomers and symmetry adapted perturbation theory for the interaction energy (DFT-SAPT). The potential is fitted in terms of analytic functions, and the fitted potential is used to compute the lowest rovibrational states of the dimer. The quality of the PES is comparable to that of a previously published surface [G. W. M. Vissers, P. E. S. Wormer, and A. van der Avoird, Phys. Chem. Chem. Phys., 5, 4767 (2003)], which was calculated with the coupled cluster single double and perturbative triples [CCSD(T)] method. It is shown that a weighted average of the DFT-SAPT and the CCSD(T) potential gives results that are in very good agreement with experimental data, for both (12CO)2 and (13CO)2 . The relative weight was determined by adjusting the energy gap between the origins of the lowest two stacks of rotational levels of (12CO)2 to the measured value.