Experiments were performed to determine the sulfur solubilities of hydrous basalts from Vesuvius, Etna and Stromboli (Italy). The melts were equilibrated at 1050 and 1200 degrees C with H2O and sulfur (added as pyrrhotite), and at pressures ranging from 250 to 2000 bar. Most experiments were performed under oxidising conditions (NNO + 2), and a few under reducing conditions (NNO - 1), with melt water contents of 0.5-3.5 wt.%. Sulfur contents in glasses were determined by electron microprobe and range from 860 up to 6700 ppm. No compositional effect is found between the three alkali basaltic melts. The fugacities of S-bearing species were derived using an MRK equation of state applied to an O-H-S fluid, knowing H-2 and H2O fugacities, and range from 50 up to 3000 bar. A thermodynamic species-based model is derived from our results along with available data in the literature, assuming that sulfur dissolution results from the additive contributions of both H2S and SO2 dissolution reactions. Compared to similar models developed for silicic melts, basalt compositions requires the incorporation of an Fe term, which accounts for the strong association between Fe and S in silicate melts, and considers the elevated Fe content of mafic melts. The model shows that, at any fixed fS(2), the sulfur solubility in hydrous basalt displays a pronounced minimum around NNO, the position of which depends on temperature. The minimum in sulfur solubility coincides with the redox range were the abundance of S-2 in the fluid reaches its maximum compared to either H2S or SO2 species. Such a minimum in solubility is in agreement with experimental constraints at 1 bar under carefully controlled fO(2) and fS(2). Calculated proportions of dissolved species in the melt depend on the prevailing fS(2) and fO(2), being in general agreement with available spectroscopic models. Calculations of gas saturation pressures, which classically consider only H2O and CO2 dissolved volatiles, are strongly affected by S-bearing species. At fO(2) close to, or higher than, NNO + 1, omission of sulfur species may result in underestimates of gas saturation pressures of 1 kbar or more. The same happens at fO(2) below NNO - 1.