Alloys from ZnO and ZnS have been synthesized by radio-frequency magnetron sputtering over the entire alloying range. The ZnO1-xSx films are crystalline for all compositions. The optical absorption edge of these alloys decreases rapidly with small amount of added sulfur (x=0.02) and continues to red shift to a minimum of 2.6 eV at x=0.45. At higher sulfur concentrations (x>0.45), the absorption edge shows a continuous blue shift. The strong reduction in the band gap for O-rich alloys is the result of the upward shift of the valence-band edge with x as observed by x-ray photoelectron spectroscopy. As a result, the room temperature bandgap of ZnO1-xSx alloys can be tuned from 3.7 eV to 2.6 eV. The observed large bowing in the composition dependence of the energy bandgap arises from the anticrossing interactions between (1) the valence-band of ZnO and the localized sulfur level at 0.30 eV above the ZnO valence-band maximum for O-rich alloys and (2) the conduction-band of ZnS and the localized oxygen level at 0.20 eV below the ZnS conduction band minimum for the S-rich alloys. The ability to tune the bandgap and knowledge of the location of the valence and conduction-band can be advantageous in applications, such as heterojunction solar cells, where band alignment is crucial.