Rising seawater temperature and ocean acidification threaten the survival of coral reefs. The relationship between coral physiology and its microbiome may reveal why some corals are more resilient to these global change conditions. Here, we conducted the first experiment to simultaneously investigate changes in the coral microbiome and coral physiology in response to the dual stress of elevated seawater temperature and ocean acidification expected by the end of this century. Two species of corals, Acropora millepora containing the thermally sensitive endosymbiont C21a and Turbinaria reniformis containing the thermally tolerant endosymbiont Symbiodinium trenchi, were exposed to control (26.5°C and pCO2 of 364 μatm) and treatment (29.0°C and pCO2 of 750 μatm) conditions for 24 days, after which we measured the microbial community composition. These microbial findings were interpreted within the context of previously published physiological measurements from the exact same corals in this study (calcification, organic carbon flux, ratio of photosynthesis to respiration, photosystem II maximal efficiency, total lipids, soluble animal protein, soluble animal carbohydrates, soluble algal protein, soluble algal carbohydrate, biomass, endosymbiotic algal density, and chlorophyll a). Overall, dually stressed A. millepora had reduced microbial diversity, experienced large changes in microbial community composition, and experienced dramatic physiological declines in calcification, photosystem II maximal efficiency, and algal carbohydrates. In contrast, the dually stressed coral T. reniformis experienced a stable and more diverse microbiome community with minimal physiological decline, coupled with very high total energy reserves and particulate organic carbon release rates. Thus, the microbiome changed and microbial diversity decreased in the physiologically sensitive coral with the thermally sensitive endosymbiotic algae but not in the physiologically tolerant coral with the thermally tolerant endosymbiont. Our results confirm recent findings that temperature-stress tolerant corals have a more stable microbiome, and demonstrate for the first time that this is also the case under the dual stresses of ocean warming and acidification. We propose that coral with a stable microbiome are also more physiologically resilient and thus more likely to persist in the future, and shape the coral species diversity of future reef ecosystems.