Atacama Large Millimeter/submillimeter Array (ALMA) 12CO(J=1-0) observations are used to study the cold molecular ISM of the Cartwheel ring galaxy and its relation to HI and massive star formation (SF). CO moment maps find $(2.69pm0.05)times10^{9}$ M$_{odot}$ of H$_2$ associated with the inner ring (72%) and nucleus (28%) for a Galactic I(CO)-to-N(H2) conversion factor ($alpha_{rm CO}$). The spokes and disk are not detected. Analysis of the inner rings CO kinematics show it to be expanding ($V_{rm exp}=68.9pm4.9$ km s$^{-1}$) implying an $approx70$ Myr age. Stack averaging reveals CO emission in the starburst outer ring for the first time, but only where HI surface density ($Sigma_{rm HI}$) is high, representing $M_{rm H_2}=(7.5pm0.8)times10^{8}$ M$_{odot}$ for a metallicity appropriate $alpha_{rm CO}$, giving small $Sigma_{rm H_2}$ ($3.7$ M$_{odot}$ pc$^{-2}$), molecular fraction ($f_{rm mol}=0.10$), and H$_2$ depletion timescales ($tau_{rm mol} approx50-600$ Myr). Elsewhere in the outer ring $Sigma_{rm H_2}lesssim 2$ M$_{odot}$ pc$^{-2}$, $f_{rm mol}lesssim 0.1$ and $tau_{rm mol}lesssim 140-540$ Myr (all $3sigma$). The inner ring and nucleus are H$_2$-dominated and are consistent with local spiral SF laws. $Sigma_{rm SFR}$ in the outer ring appears independent of $Sigma_{rm H_2}$, $Sigma_{rm HI}$ or $Sigma_{rm HI+H_2}$. The ISMs long confinement in the robustly star forming rings of the Cartwheel and AM0644-741 may result in either a large diffuse H$_2$ component or an abundance of CO-faint low column density molecular clouds. The H$_2$ content of evolved starburst rings may therefore be substantially larger. Due to its lower $Sigma_{rm SFR}$ and age the Cartwheels inner ring has yet to reach this state. Alternately, the outer ring may trigger efficient SF in an HI-dominated ISM.
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