A study of electron densities in Saturn's inner magnetosphere is presented using measurements of the upper hybrid resonance frequency obtained from the Radio and Plasma Wave Science (RPWS) instrument on the Cassini spacecraft. The study uses data from the first 16 months of operation in orbit around Saturn. The distribution of density data spans latitudes up to 20¿ and ranges from 3.6 ≤ L ≤ 8.6. The results are compared to a simple centrifugal potential model for the plasma density. The measurements for 5 ≤ L ≤ 8.6 yield a good fit to an equatorial electron density profile that varies as n0 = (51,880) L-4.1 cm-3 and a plasma scale height that varies as H = (0.047) L1.8 RS, where RS is the radius of Saturn. The measurements for L < 5 are more variable, most likely due to plasma injection effects by Saturn's moon Enceladus which is a known source of neutral gas. A contour map of the electron densities derived from the centrifugal potential model is presented over the entire range of L values and latitudes analyzed, using cubic polynomials to represent the radial profiles of the equatorial electron density and the plasma scale height.