ZnS1-xOx is a highly mismatched semiconductor alloy with potential light-emitting and solar-cell applications. In this work, optical transmission spectroscopy and a modified derivative method were employed to determine the room-temperature bandgap of cubic (zinc blende) ZnS1-xOx from x = 0.01 to 0.3. The bandgap drops steeply for dilute oxygen concentrations, followed by a more gradual decrease for x > 0.05. This nonlinear behavior is attributed to a transition from isolated oxygen impurities to pairs and larger clusters. Alloying with x = 0.3 causes bandgap to drop from 3.7 to 3.1 eV. Previous work showed that the bandgap of wurtzite ZnS1-xOx shifts from 3.7 to 2.8 eV over the same composition range.