We report on the discovery of a planetary system with a close-in transiting hot Jupiter on a near circular orbit and a massive outer planet on a highly eccentric orbit. The inner planet, HAT-P-13b, transits the bright V=10.622 G4 dwarf star GSC 3416-00543 every P = 2.916260 ± 0.000010 days, with transit epoch Tc = 2454779.92979 ± 0.00038 (BJD) and duration 0.1345 ± 0.0017 d. The outer planet, HAT-P-13c orbits the star with P2 = 428.5 ± 3.0 days and nominal transit center (assuming zero impact parameter) of T2c = 2454870.4 ± 1.8 (BJD) or time of periastron passage T2,peri= 2454890.05 ± 0.48 (BJD). Transits of the outer planet have not been observed, and may not be present. The host star has a mass of 1.22 ± ^0.05_0.10 Msun, radius of 1.56 ± 0.08 Rsun, effective temperature 5653 ± 90 K, and is rather metal rich with [Fe=H] = +0.41 ± 0.08. The inner planetary companion has a mass of 0.853± ^0.029_-0.046MJup, and radius of 1.281 ± 0.079 RJup yielding a mean density of 0.498± +0.103_-0.069 gcm^-3. The outer companion has m2 sini2 = 15.2 ± 1.0 MJup, and orbits on a highly eccentric orbit of e2 = 0.691 ± 0.018. While we have not detected significant transit timing variations of HAT-P-13b, due to gravitational and light-travel time effects, future observations will constrain the orbital inclination of HAT-P-13c, along with its mutual inclination to HAT-P-13b. The HAT-P-13 (b,c) double-planet system may prove extremely valuable for theoretical studies of the formation and dynamics of planetary systems. ; Comment: Accepted for publication in ApJ. 11 pages, 7 figures