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 \pm 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 \approx 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 \gtrsim 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 } \gg 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 .