[abridged] We report four years of radial velocity monitoring observations of SDSS J080531.84+481233.0 that reveal significant and periodic variability, confirming the binary nature of the source. We infer an orbital period of 2.02$pm$0.03 yr, a semi-major axis of 0.76$^{+0.05}_{-0.06}$ AU, and an eccentricity of 0.46$pm$0.05, consistent with the amplitude of astrometric variability and prior attempts to resolve the system. Folding in constraints based on the spectral types of the components (L4$pm$0.7 and T5.5$pm$1.1), corresponding effective temperatures, and brown dwarf evolutionary models, we further constrain the orbital inclination of this system to be nearly edge-on (90$^opm$19$^o$), and deduce a large system mass ratio (M$_2$/M$_1$ = 0.86$^{+0.10}_{-0.12}$), substellar components (M$_1$ = 0.057$^{+0.016}_{-0.014}$ M$_{odot}$, M$_2$ = 0.048$^{+0.008}_{-0.010}$ M$_{odot}$), and a relatively old system age (minimum age = 4.0$^{+1.9}_{-1.2}$ Gyr). The measured projected rotational velocity of the primary ($vsin{i}$ = 34.1$pm$0.7 km/s) implies that this inactive source is a rapid rotator (period $lesssim$ 3 hr) and a viable system for testing spin-orbit alignment in very-low-mass multiples. The combination of well-determined component atmospheric properties and masses near and/or below the hydrogen minimum mass make SDSS J0805+4812AB an important system for future tests of brown dwarf evolutionary models.
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