We report on the discovery of HAT-P-12b, a transiting extrasolar planet orbiting the moderately bright V 12.8 K4 dwarf GSC 03033 - 00706, with a period P = 3.2130598 +/- 0.0000021 d, transit epoch Tc = 2454419.19556 +/- 0.00020 (BJD), and transit duration 0.0974 +/- 0.0006 d. The host star has a mass of 0.73 +/- 0.02 M, radius of 0.70+0.02 -0.01 R, effective temperature 4650 +/- 60 K, and metallicity Fe/H = -0.29 +/- 0.05. We find a slight correlation between the observed spectral line bisector spans and the radial velocity, so we consider, and rule out, various blend configurations including a blend with a background eclipsing binary, and hierarchical triple systems where the eclipsing body is a star or a planet. We conclude that a model consisting of a single star with a transiting planet best fits the observations, and show that a likely explanation for the apparent correlation is contamination from scattered moonlight. Based on this model, the planetary companion has a mass of 0.211 +/- 0.012 M J and radius of 0.959+0.029 -0.021 R J yielding a mean density of 0.295 +/- 0.025 g cm-3. Comparing these observations with recent theoretical models, we find that HAT-P-12b is consistent with a ~1-4.5 Gyr, mildly irradiated, H/He-dominated planet with a core mass MC 10 M {circled plus}. HAT-P-12b is thus the least massive H/He-dominated gas giant planet found to date. This record was previously held by Saturn.