We present Keck laser guide star adaptive optics imaging of the M8+M8 binary 2MASS J2206 – 2047AB. Together with archival Hubble Space Telescope, Gemini-North, and Very Large Telescope data, our observations span 8.3 yr of the binary's 35⁺⁶ _–5 yr orbital period, and we determine a total dynamical mass of 0.15^+0.05 _–0.03 M _☉, with the uncertainty dominated by the parallax error. Using the measured total mass and individual luminosities, the Tucson and Lyon evolutionary models both give an age for the system of 0.4^+9.6 _–0.2 Gyr, which is consistent with its thin disk space motion derived from the Besançon Galactic structure model. Our mass measurement combined with the Tucson (Lyon) evolutionary models also yields precise effective temperatures, giving 2660⁺⁹⁰ _–100 K and 2640⁺⁹⁰ _–100 K (2550⁺⁹⁰ _–100 K and 2530⁺⁹⁰ _–100 K) for components A and B, respectively. These temperatures are in good agreement with estimates for other M8 dwarfs (from the infrared flux method and the M8 mass benchmark LHS 2397aA), but atmospheric model fitting of the integrated-light spectrum gives hotter temperatures of 2800 ± 100 K for both components. This modest discrepancy can be explained by systematic errors in the atmospheric models or by a slight underestimation of the distance (and thus, mass and age) of the system. We also find that the observed near-infrared colors and magnitudes do not agree with those predicted by the Lyon Dusty models, given the known mass of the system.