The Orbit of the L dwarf + T dwarf Spectral Binary SDSS J080531.84+481233.0

[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$1.1), corresponding effective temperatures, and brown dwarf evolutionary models, we further constrain the orbital inclination of this system to be nearly edge-on (90$^o\pm$19$^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 ($v\sin{i}$ = 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.