The cloning of agronomically important genes from large, complex crop genomes remains challenging. Here we generate a 14.7 gigabase chromosome-scale assembly of the South African bread wheat (Triticum aestivum) cultivar Kariega by combining high-fidelity long reads, optical mapping and chromosome conformation capture. The resulting assembly is an order of magnitude more contiguous than previous wheat assemblies. Kariega shows durable resistance to the devastating fungal stripe rust disease. We identified the race-specific disease resistance gene Yr27, which encodes an intracellular immune receptor, to be a major contributor to this resistance. Yr27 is allelic to the leaf rust resistance gene Lr13; the Yr27 and Lr13 proteins show 97% sequence identity. Our results demonstrate the feasibility of generating chromosome-scale wheat assemblies to clone genes, and exemplify that highly similar alleles of a single-copy gene can confer resistance to different pathogens, which might provide a basis for engineering Yr27 alleles with multiple recognition specificities in the future.