A major challenge in quantum optics and quantum information technology is to enhance the interaction between single photons and single quantum emitters. This requires highly engineered optical cavities that are inherently sensitive to fabrication imperfections. We have demonstrated a fundamentally different approach in which disorder is used as a resource rather than a nuisance. We generated strongly confined Anderson-localized cavity modes by deliberately adding disorder to photonic crystal waveguides. The emission rate of a semiconductor quantum dot embedded in the waveguide was enhanced by a factor of 15 on resonance with the Anderson-localized mode, and 94% of the emitted single photons coupled to the mode. Disordered photonic media thus provide an efficient platform for quantum electrodynamics, offering an approach to inherently disorder-robust quantum information devices.