Antibiotics can induce mutations that cause antibiotic resistance. Yet, despite their importance, mechanisms of antibiotic-promoted mutagenesis remain elusive. We report that the fluoroquinolone antibiotic ciprofloxacin (cipro) induces mutations that cause drug resistance by triggering differentiation of a mutant-generating cell subpopulation, using reactive oxygen species (ROS) to signal the sigma-S (σS) general-stress response. Cipro-generated DNA breaks activate the SOS DNA-damage response and error-prone DNA polymerases in all cells. However, mutagenesis is restricted to a cell subpopulation in which electron transfer and SOS induce ROS, which activate the σS response, allowing mutagenesis during DNA-break repair. When sorted, this small σS-response-'on' subpopulation produces most antibiotic cross-resistant mutants. An FDA-approved drug prevents σS induction specifically inhibiting antibiotic-promoted mutagenesis. Furthermore, SOS-inhibited cell division, causing multi-chromosome cells, is required for mutagenesis. The data support a model in which within-cell chromosome cooperation together with development of a 'gambler' cell subpopulation promote resistance evolution without risking most cells.