Given climate change, thermal stress-related mass coral-bleaching events present one of the greatest anthropogenic threats to coral reefs. While corals and their symbiotic algae may respond to future temperatures through genetic adaptation and shifts in community compositions, the climate may change too rapidly for coral response. To test this potential for response, here we develop a model of coral and symbiont ecological dynamics and symbiont evolutionary dynamics. Model results without variation in symbiont thermal tolerance predict coral reef collapse within decades under multiple future climate scenarios, consistent with previous threshold-based predictions. However, model results with genetic or community-level variation in symbiont thermal tolerance can predict coral reef persistence into the next century, provided low enough greenhouse gas emissions occur. Therefore, the level of greenhouse gas emissions will have a significant effect on the future of coral reefs, and accounting for biodiversity and biological dynamics is vital to estimating the size of this effect.