  Rate coefficients for state-to-state rotational transitions in CO induced by
both para- and ortho-H$_2$ collisions are presented. The results were obtained
using the close-coupling method and the coupled-states approximation, with the
CO-H$_2$ interaction potential of Jankowski & Szalewicz (2005). 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 $j^\prime$ levels. Comparisons
with previous calculations using an earlier potential show some discrepancies,
especially at low temperatures and for rotational transitions involving large
$|\Delta j|$. The differences in the well depths of the van der Waals
interactions in 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.
