We present Keck laser guide star adaptive optics imaging and aperture masking observations of the M8+L7binary LHS2397aAB. Together with archival Hubble Space Telescope, Gemini-North, and Very Large Telescope data, our observations span 11.8years of the binary's 14.2year orbital period. We determine a total dynamical mass of 0.146+0.015-0.013M ⊙ (153+16-14 M Jup). Using the combined observational constraints of the total mass and individual luminosities, the Tucson (Lyon) evolutionary models give an age for the system of 1.5+4.1-0.6 Gyr (1.8+8.2-0.8 Gyr), which is consistent with its space motion based on a comparison to the Besançon Galactic structure model. We also use these models to determine the mass ratio, giving individual masses of 0.0839+0.0007-0.0015 M ȯ (0.0848+0.0010-0.0012M ȯ) for LHS2397aA and 0.061+0.014-0.011M ȯ (0.060+0.008-0.012M ȯ) for LHS2397aB. Because LHS2397aB is very close to the theoretical mass limit of lithium burning, which remains untested by dynamical masses, measuring its lithium depletion would uniquely test substellar models. We estimate a spectral type of L7 1for LHS2397aB, making it the first L/T transition object with a dynamical mass determination. This enables a precise estimate of its effective temperature from Tucson (Lyon) evolutionary models of 1450 40 K (1430 40 K), which is 200 K higher than estimates for young late-L companions but consistent with older late-L field dwarfs, supporting the idea that the temperature of t he L/T transition is surface gravity dependent. Comparing our temperature estimate for LHS2397aB to those derived from spectral synthesis modeling for similar objects reveals consistency between evolutionary and atmospheric models at the L/T transition, despite the currently limited understanding of this phase of substellar evolution. Future dynamical masses for L/T binaries spanning a range of surface gravity, age, and mass will provide the next critical tests of substellar models at the L/T transition.