We combine data from the Australia Telescope National Facility and the Swedish ESO Submillimeter Telescope to investigate the neutral interstellar medium (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 0.079 ± 0.005. Apart from the starburst peak, the gas ring appears stable against the growth of gravitational instabilities (Q gas = 2 − 7). Including the stellar component lowers Q overall, but not enough to make Q < 1 everywhere. The ring's global star formation efficiency (SFE) appears somewhat elevated relative to early spirals, 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 H 2 is uncorrelated with star formation rate density. 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 amplifies 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 M H 2 to be underestimated by 12 CO 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.3 Visiting astronomer, European Southern Observatory. 4 Visiting astronomer, Cerro Tololo Inter-American Observatory.properties of the star-forming ISM in an evolved ring galaxy compare with other systems?This paper is organized as follows: The observations and data reduction are outlined in Section 2. The results derived using these observations are presented in Section 3, including the ring's metallicity, the distribution and kinematics of HI, the relative proportions of molecular and atomic gas, star formation efficiency, and their variations around the ring. We also consider the ring's gravitational stability, and whether or not star formation follows a Schmidt law. In Section 4 we consider the role played by pressure and ambient FUV radiation field in determining the ring's molecular fraction. We also discuss our results in the context of an alternate view of ring galaxies, in which the ring represents a self-sustaining propagating starburst. Our conclusions are given in Section 5. Throughout this paper we adopt a flat ΛCDM cosmology based on the 7 year WMAP results (Komatsu et al. 2011),...