Although reef-building corals are declining worldwide, responses to bleaching vary within and across species and are partly heritable. Toward predicting bleaching response from genomic data, we generated a chromosome-scale genome assembly for the coral Acropora millepora. We obtained whole-genome sequences for 237 phenotyped samples collected at 12 reefs along the Great Barrier Reef, among which we inferred little population structure. Scanning the genome for evidence of local adaptation, we detected signatures of long-term balancing selection in the heat-shock co-chaperone sacsin. We conducted a genome-wide association study of visual bleaching score for 213 samples, incorporating the polygenic score derived from it into a predictive model for bleaching in the wild. These results set the stage for genomics-based approaches in conservation strategies.