Rate coefficients for state-to-state rotational transitions in CO induced by both para- and ortho-H2 collisions are presented. The results were obtained using the close-coupling method and the coupled-states approximation, with the CO-H2 interaction potential of Jankowski & Szalewicz. Rate coefficients are presented for temperatures between 1 and 3000 K, and for CO(v = 0, j) quenching from j = 1-40 to all lower j' levels. Comparisons with previous calculations using an earlier potential show some discrepancies, especially at low temperatures and for rotational transitions involving large |Δj|. The differences in the well depths of the van der Waals interactions and the anisotropy of the two potential surfaces lead to different resonance structures in the energy dependence of the cross sections which influence the low temperature rate coefficients. Applications to far infrared observations of astrophysical environments are briefly discussed.