We report the discovery by the HATSouth network of HATS-7b, a transiting Super-Neptune with a mass of 0.120 +/- 0.012 M-J, a radius of 0.563(-0.034)(+0.046) R-J, and an orbital period of 3.1853 days. The host star is a moderately bright (V = 13.340 +/- 0.010 mag, K-S = 10.976 +/- 0.026 mag) K dwarf star with a mass of 0.849 +/- 0.027 M-circle dot, a radius of 0.815(-0.035)(+0.049) R-circle dot, and a metallicity of [Fe/H]= +0.250 +/- 0.080. The star is photometrically quiet to within the precision of the HATSouth measurements, has low RV jitter, and shows no evidence for chromospheric activity in its spectrum. HATS-7b is the second smallest radius planet discovered by a wide-field ground-based transit survey, and one of only a handful of Neptune-size planets with mass and radius determined to 10% precision. Theoretical modeling of HATS-7b yields a hydrogen-helium fraction of 18 +/- 4% (rock-iron core and H-2-He envelope), or 9 +/- 4% (ice core and H-2-He envelope), i.e., it has a composition broadly similar to that of Uranus and Neptune, and very different from that of Saturn, which has 75% of its mass in H-2-He. Based on a sample of transiting exoplanets with accurately (<20%) determined parameters, we establish approximate power-law relations for the envelopes of the mass-density distribution of exoplanets. HATS-7b, which, together with the recently discovered HATS-8b, is one of the first two transiting super-Neptunes discovered in the Southern sky, is a prime target for additional follow-up observations with Southern hemisphere facilities to characterize the atmospheres of Super-Neptunes (which we define as objects with mass greater than that of Neptune, and smaller than halfway between that of Neptune and Saturn, i.e., 0.054M(J) < M-p < 0.18M(J)).