Cross sections for the rotational (de)excitation of CO by ground state para-and ortho-H 2 are obtained using quantum scattering calculations for collision energies between 1 and 520 cm −1 . A new CO−H 2 potential energy surface is employed and its quality is assessed by comparison with explicitly correlated CCSD(T)-R12 calculations. Rate constants for rotational levels of CO up to 5 and temperatures in the range 5−70 K are deduced. The new potential is found to have a strong influence on the resonance structure of the cross sections at very low collision energies. As a result, the present rates at 10 K differ by up to 50% with those obtained by Flower (2001) on a previous, less accurate, potential energy surface.