The possibility that snowmelt could have provided liquid water for valley network formation early in the history of Mars is investigated using an optical-thermal model developed for dusty snowpacks at temperate latitudes. The heating of the postulated snow is assumed to be driven primarily by the absorption of solar radiation during clear sky conditions. Radiative heating rates are predicted as a function of depth and shown to be sensitive to the dust concentration and the size of the ice grains while the thermal conductivity is controlled by temperature, atmospheric pressure, and bulk density. Rates of metamorphism indicate that fresh fine-grained snow on Mars would evolve into moderately coarse snow during a single summer season. Results from global climate models are used to constrain the mean-annual surface temperatures for snow and the atmospheric exchange terms in the surface energy balance. Mean-annual temperatures within Martian snowpacks fail to reach the melting point for all atmospheric pressures below 1000 mbar despite a predicted temperature enhancement beneath the surface of the snowpacks. When seasonal and diurnal variations in the incident solar flux are included in the model, melting occurs at midday during the summer for a wide range of snow types and atmospheric pressures if the dust levels in the snow exceed 100 ppmw (parts per million by weight). The optimum dust concentration appears to be about 1000 ppmw. With this dust load, melting can occur in the upper few centimeters of a dense coarse-grained snow at atmospheric pressures as low as 7 mbar. Snowpack thickness and the thermal conductivity of the underlying substrate determine whether the generated snow-melt can penetrate to the snowpack base, survive basal ice formation, and subsequently become available for runoff. Under favorable conditions, liquid water becomes available for runoff at atmospheric pressures as low as 30 to 100 mbar if the substrate is composed of regolith, as is expected in the ancient cratered terrain of Mars.