We present the Hubble Space Telescope (HST) Space Telescope Imaging Spectrograph (STIS) optical transmission spectroscopy of the cool Saturn-mass exoplanet WASP-39b from 0.29-1.025 μm, along with complementary transit observations from Spitzer IRAC at 3.6 and 4.5 μm. The low density and large atmospheric pressure scale height of WASP-39b make it particularly amenable to atmospheric characterization using this technique. We detect a Rayleigh scattering slope as well as sodium and potassium absorption features; this is the first exoplanet in which both alkali features are clearly detected with the extended wings predicted by cloud-free atmosphere models. The full transmission spectrum is well matched by a clear H₂-dominated atmosphere, or one containing a weak contribution from haze, in good agreement with the preliminary reduction of these data presented in Sing et al. WASP-39b is predicted to have a pressure-temperature profile comparable to that of HD 189733b and WASP-6b, making it one of the coolest transiting gas giants observed in our HST STIS survey. Despite this similarity, WASP-39b appears to be largely cloud-free, while the transmission spectra of HD 189733b and WASP-6b both indicate the presence of high altitude clouds or hazes. These observations further emphasize the surprising diversity of cloudy and cloud-free gas giant planets in short-period orbits and the corresponding challenges associated with developing predictive cloud models for these atmospheres.