astroDDPM uses a denoising diffusion probabilistic model (DDPM) to synthesize galaxies that are qualitatively and physically indistinguishable from the real thing. The similarity of the synthesized images to real galaxies from the Photometry and Rotation curve OBservations from Extragalactic Surveys (PROBES) sample and from the Sloan Digital Sky Survey is quantified using the Fréchet Inception Distance to test for subjective and morphological similarity. The emergent physical properties (such as total magnitude, color, and half light radius) of a ground truth parent and synthesized child dataset are also compared to generate a Synthetic Galaxy Distance metric. The DDPM approach produces sharper and more realistic images than other generative methods such as Adversarial Networks (with the downside of more costly inference), and could be used to produce large samples of synthetic observations tailored to a specific imaging survey. Potential uses of the DDPM include accurate in-painting of occluded data, such as satellite trails, and domain transfer, where new input images can be processed to mimic the properties of the DDPM training set.