The satellites Rhea and Dione orbit within the magnetosphere of Saturn, where they are exposed to particle irradiation from trapped ions. A similar situation applies to the galilean moons Europa, Ganymede and Callisto, which reside within Jupiter's radiation belts. All of these satellites have surfaces rich in water ice. Laboratory studies of the interaction of charged-particle radiation with water ice predicted the tenuous oxygen atmospheres recently found on Europa and Ganymede. However, theoretical investigations did not anticipate the trapping of significantly larger quantities of O2 within the surface ice. The accumulation of detectable abundances of O3, produced by the action of ultraviolet or charged-particle radiation on O2, was also not predicted before being observed on Ganymede. Here we report the identification of O3 in spectra of the saturnian satellites Rhea and Dione. The presence of trapped O3 is thus no longer unique to Ganymede, suggesting that special circumstances may not be required for its production.