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+/-0.0000021d, transit epoch T_c_=2454419.19556+/-0.00020 (BJD), and transit duration 0.0974+/-0.0006d. The host star has a mass of 0.73+/-0.02M_{sun}, radius of 0.70^+0.02^-0.01_R_{sun}, effective temperature 4650+/-60K, 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+/-0.012M_J and radius of 0.959^+0.029^-0.021_R_J yielding a mean density of 0.295+/-0.025g/cm^3^. Comparing these observations with recent theoretical models, we find that HAT-P-12b is consistent with a ~1-4.5Gyr, mildly irradiated, H/He-dominated planet with a core mass M_C_<~10M_{earth}_.