Helicobacter pylori infection is a major risk factor for the development of gastric cancer. The bacteria reside in close proximity to gastric surface mucous as well as stem and progenitor cells. Here, we take advantage of wild-type and genetically engineered murine gastric organoids and organoid-derived monolayers to study the cellular targets of H. pylori-induced DNA damage and replication stress and to explore possible interactions with preexisting gastric cancer driver mutations. We find using alkaline comet assay, single-molecule DNA fiber assays, and immunofluorescence microscopy of DNA repair foci that H. pylori induces transcription-dependent DNA damage in actively replicating, Leucine-rich-repeat containing G-Protein-Coupled Receptor 5 (Lgr5)-positive antral stem and progenitor cells and their Troy-positive corpus counterparts, but not in other gastric epithelial lineages. Infection-dependent DNA damage is aggravated by Apc inactivation, but not by Trp53 or Smad4 loss, or Erbb2 overexpression. Our data suggest that H. pylori induces DNA damage in stem and progenitor cells, especially in settings of hyperproliferation due to constitutively active Wnt signaling.