The process by which massive galaxies transition from blue, star-forming disks into red, quiescent galaxies remains one of the most poorly understood aspects of galaxy evolution. In this investigation, we attempt to gain a better understanding of how star formation is quenched by focusing on a massive post-starburst galaxy at z = 0.747. The target has a high stellar mass and a molecular gas fraction of 30%—unusually high for its low star formation rate (SFR). We look for indicators of star formation suppression mechanisms in the stellar kinematics and age distribution of the galaxy obtained from spatially resolved Gemini Integral-field spectra and in the gas kinematics obtained from the Atacama Large Millimeter/submillimeter Array (ALMA). We find evidence of significant rotation in the stars, but we do not detect a stellar age gradient within 5 kpc. The molecular gas is aligned with the stellar component, and we see no evidence of strong gas outflows. Our target may represent the product of a merger-induced starburst or of morphological quenching; however, our results are not completely consistent with any of the prominent quenching models.