We report the discovery of an L-type companion to the young M3.5V star 2MASS J01225093-2439505 at a projected separation of 1. 45 (≈52 AU) as part of our adaptive optics imaging search for extrasolar giant planets around young low-mass stars. 2MASS 0122-2439 B has very red near-infrared colors similar to the HR 8799 planets and the reddest known young/dusty L dwarfs in the field. Moderate-resolution (R ≈ 3800) 1.5-2.4 μm spectroscopy reveals a near-infrared spectral type of L4-L6 and an angular H-band shape, confirming its cool temperature and young age. The kinematics of 2MASS 0122-2439 AB are marginally consistent with members of the ∼120 Myr AB Dor young moving group based on the photometric distance to the primary (36 ± 4 pc) and our radial velocity measurement of 2MASS 0122-2439 A from Keck/HIRES. We adopt the AB Dor group age for the system, but the high energy emission, lack of Li i λ6707 absorption, and spectral shape of 2MASS 0122-2439 B suggest a range of ∼10-120 Myr is possible. The age and luminosity of 2MASS 0122-2439 B fall in a strip where "hot-start" evolutionary model mass tracks overlap as a result of deuterium burning. Several known substellar companions also fall in this region (2MASS J0103-5515 ABb, AB Pic b, κ And b, G196-3 B, SDSS 2249+0044 B, LP 261-75 B, HD 203030 B, and HN Peg B), but their dual-valued mass predictions have largely been unrecognized. The implied mass of 2MASS 0122-2439 B is ≈12-13 M Jup or ≈22-27 M Jup if it is an AB Dor member, or possibly as low as 11 M Jup if the wider age range is adopted. Evolutionary models predict an effective temperature for 2MASS 0122-2439 B that corresponds to spectral types near the L/T transition (≈1300-1500 K) for field objects. However, we find a mid-L near-infrared spectral type, indicating that 2MASS 0122-2439 B represents another case of photospheric dust being retained to cooler temperatures at low surface gravities, as seen in the spectra of young (8-30 Myr) planetary companions. Altogether, the low mass, low temperature, and red colors of 2MASS 0122-2439 B make it a bridge between warm planets like β Pic b and cool, very dusty ones like HR 8799 bcde.