During the two Voyager encounters with Saturn, radio bursts were detected that appear to have originated from atmospheric lightning storms. Although these bursts generally extended over frequencies from as low as 100 kHz to the upper detection limit of the instrument, 40 MHz, they often exhibited a sharp but variable low-frequency cutoff below which bursts were not detected. We interpret the variable low-frequency extent of these bursts to be due to the refraction and reflection of the radio waves as they propagate through an ionosphere that varies with local time. We obtain estimates of electron densities at a variety of latitude and local time locations. These compare well with the dawn and dusk densities measured by the Pioneer 11 and Voyager radio science investigations and with model predictions for dayside densities. However, we infer a two order of magntiude diurnal variation of electron densities, which had not been anticipated by theoretical models of Saturn's ionosphere, and an equally dramatic extinction of ionospheric electron density by Saturn's rings.