  We combine data from the ATNF and the SEST to investigate the neutral ISM in
AM0644-741, a large and robustly star-forming ring galaxy. The galaxy's ISM is
concentrated in the 42-kpc diameter starburst ring, but appears dominated by
atomic gas, with a global molecular fraction (f_mol) of only 7.9%. Apart from
the starburst peak, the gas ring is stable against the growth of gravitational
instabilities (Q_gas=2-7). Including stars lowers Q overall, but not enough to
make Q<1 everywhere. The ring's global star formation efficiency (SFE) appears
somewhat elevated, but varies around the ring by more than an order of
magnitude, peaking where star formation is most intense. AM0644-741's star
formation law is peculiar: HI follows a Schmidt law while H2 is uncorrelated
with SFR/area. Photodissociation models yield low volume densities in the ring,
particularly in the starburst quadrant (n~2 cm^-3), implying a warm neutral
medium dominated ISM. At the same time, the ring's pressure and ambient
far-ultraviolet radiation field lead to the expectation of a predominantly
molecular ISM. We argue that the ring's peculiar star formation law, n, SFE,
and f_mol result from the ISM's >100 Myr confinement time in the starburst
ring, which enhances the destructive effects of embedded massive stars and
supernovae. As a result, the ring's molecular ISM becomes dominated by small
clouds where star formation is most intense, causing H2 to be underestimated by
12CO line fluxes: in effect X(CO) >> X(Gal) despite the ring's solar
metallicity. The observed large HI component is primarily a low density
photodissociation product, i.e., a tracer rather than a precursor of massive
star formation. Such an "over-cooked" ISM may be a general characteristic of
evolved starburst ring galaxies.
