Plastids perform crucial cellular functions, including photosynthesis, across a wide variety of eukaryotes. Since endosymbiosis, plastids have maintained independent genomes that now display a wide diversity of gene content, genome structure, gene regulation mechanisms, and transmission modes. The evolution of plastid genomes depends on an input of de novo mutation, but our knowledge of mutation in the plastid is limited to indirect inference from patterns of DNA divergence between species. Here, we use a mutation accumulation experiment, where selection acting on mutations is rendered ineffective, combined with whole-plastid genome sequencing to directly characterize de novo mutation in Chlamydomonas reinhardtii. We show that the mutation rates of the plastid and nuclear genomes are similar, but that the base spectra of mutations differ significantly. We integrate our measure of the mutation rate with a population genomic dataset of 20 individuals, and show that the plastid genome is subject to substantially stronger genetic drift than the nuclear genome. We also show that high levels of linkage disequilibrium in the plastid genome are not due to restricted recombination, but are instead a consequence of increased genetic drift. One likely explanation for increased drift in the plastid genome is that there are stronger effects of genetic hitchhiking. The presence of recombination in the plastid is consistent with laboratory studies in C. reinhardtii and demonstrates that although the plastid genome is thought to be uniparentally inherited, it recombines in nature at a rate similar to the nuclear genome.