We report the discovery of HAT-P-25b, a transiting extrasolar planet orbiting the V = 13.19 G5 dwarf star GSC 1788-01237, with a period P = 3.652836 +/- 0.000019 days, transit epoch Tc = 2455176.85173 +/- 0.00047 (BJD), and transit duration 0.1174 +/- 0.0017 days. The host star has mass of 1.01 +/- 0.03 M(Sun), radius of 0.96 +(0.05)-(0.04) R(Sun), effective temperature 5500 +/- 80 K, and metallicity [Fe/H] = +0.31 +/- 0.08. The planetary companion has a mass of 0.567 +/- 0.022 M(Jup), and radius of 1.190 +(0.081)-(0.056) R(Jup) yielding a mean density of 0.42 +/- 0.07 g cm-3. Comparing these observations with recent theoretical models, we find that HAT-P-25b is consistent with a hydrogen-helium dominated gas giant planet with negligible core mass and age 3.2 +/- 2.3 Gyr. The properties of HAT-P-25b support several previously observed correlations for planets in the mass range 0.4 < M < 0.7 M(Jup), including those of core mass vs. metallicity, planet radius vs. equilibrium temperature, and orbital period vs. planet mass. We also note that HAT-P-25b orbits the faintest star found by HATNet to have a transiting planet to date, and is one of only a very few number of planets discovered from the ground orbiting a star fainter than V = 13.0.