We present CO observations of the H I cloud located between the Virgo elliptical galaxy NGC 4472 and its dwarf irregular companion galaxy UGC 7636. ROSAT observations of the X-ray halo of NGC 4472 show a hole in the X-ray emission at the position of the H I cloud (Irwin gi Sarazin 1996), If the hole is assumed to be the result of soft X-ray absorption by some absorbing material within the cloud, the total implied mass of the absorbing material is much larger than the measured H I mass of the cloud, This implies that a large fraction of the cloud is composed of molecular hydrogen. However, the CO observations fail to find the required amount of molecular gas if a Galactic CO-to-H-2 conversion factor is assumed. While a low density in the molecular gas might increase the CO-to-H-2 conversion factor enough to make the CO upper limit consistent with the X-ray absorption prediction too low a density would allow the H-2 to be dissociated by UV radiation. We conclude that it is unlikely that the required absorbing mass is present. In any case, these observations provide a strict upper limit on the CO content of the H I cloud. (C) 1997 American Astronomical Society.