  We have discovered a remarkable warm (130-160 K) molecular hydrogen tail with
a H_2 mass of approximately 4*10^7 solar masses extending 20 kpc from a cluster
spiral galaxy, ESO 137-001, in Abell 3627. At least half of this gas is lost
permanently to the intracluster medium, as the tail extends beyond the tidal
radius of the galaxy. We also detect a hot (400-550 K) component in the tail
that is approximately 1% of the mass. The large H_2 line to IR continuum
luminosity ratio in the tail indicates that star formation is not a major
excitation source and that the gas is possibly shock-heated. This discovery
confirms that the galaxy is currently undergoing ram-pressure stripping, as
also indicated by its X-ray and Halpha tails found previously. We estimate the
galaxy is losing its warm H_2 gas at a rate of ~ 2-3 solar masses per year. The
true mass loss rate is likely higher if we account for cold molecular gas and
atomic gas. We predict that the galaxy will lose most of its gas in a single
pass through the core and place a strong upper limit on the ram-pressure
timescale of 1 Gyr. We also study the star-forming properties of the galaxy and
its tail. We identify most of the previously discovered external Halpha sources
within the tail in our 8 um data but not in our 3.6 um data; IRS spectroscopy
of the region containing these Halpha sources also reveals aromatic features
typically associated with star formation. From the positions of these HII
regions, it appears that star formation is not occurring throughout the
molecular hydrogen tail but only immediately downstream of the galaxy. Some of
these HII regions lie outside the tidal radius of the galaxy, indicating that
ram-pressure stripping can be a source of intracluster stars.
