  [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.
