For the first time, we reveal large amounts of cold molecular gas in a ram pressure stripped tail, out to a large, intracluster distance from the galaxy. With the APEX telescope we have detected 12CO(2-1) emission corresponding to more than 10^9 Msun of H_2 in three Ha bright regions along the tail of the Norma cluster galaxy ESO 137-001, out to a projected distance of 40 kpc from the disk. ESO 137-001 has an 80 kpc long and bright X-ray tail associated with a shorter (40 kpc) and broader tail of numerous star-forming H II regions. The amount of about 1.5x10^8 Msun of H_2 found in the most distant region is similar to molecular masses of tidal dwarf galaxies, though the standard Galactic CO-to-H_2 factor could overestimate the H_2 content. Along the tail, we find the amount of molecular gas to drop, while masses of the X-ray emitting and diffuse ionized components stay roughly constant. Moreover, the amounts of hot and cold gas are large and similar, and together nearly account for the missing gas from the disk. We find a very low star formation efficiency (tau > 10^10 yr) in the stripped gas in ESO 137-001 and suggest that this is due to a low average gas density in the tail, or turbulent heating of the interstellar medium that is induced by a ram pressure shock. The unprecedented bulk of observed H_2 in the ESO 137-001 tail suggests that some stripped gas may survive ram pressure stripping in the molecular phase.
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