At a distance of 2~pc, our nearest brown dwarf neighbor, Luhman 16 AB, has been extensively studied since its discovery 3 years ago, yet its most fundamental parameter -- the masses of the individual dwarfs -- has not been constrained with precision. In this work we present the full astrometric orbit and barycentric motion of Luhman 16 AB and the first precision measurements of the individual component masses. We draw upon archival observations spanning 31 years from the European Southern Observatory (ESO) Schmidt Telescope, the Deep Near-Infrared Survey of the Southern Sky (DENIS), public FORS2 data on the Very Large Telescope (VLT), and new astrometry from the Gemini South Multiconjugate Adaptive Optics System (GeMS). Finally, we include three radial velocity measurements of the two components from VLT/CRIRES, spanning one year. With this new data sampling a full period of the orbit, we use a Markov Chain Monte Carlo algorithm to fit a 16-parameter model incorporating mutual orbit and barycentric motion parameters and constrain the individual masses to be~$27.9^{+1.1}_{-1.0}$~$M_{J}$ for the T dwarf and~$34.2^{+1.3}_{-1.1}$~$M_{J}$ for the L dwarf. Our measurements of Luhman 16 ABs mass ratio and barycentric motion parameters are consistent with previous estimates in the literature utilizing recent astrometry only. The GeMS-derived measurements of the Luhman 16 AB separation in 2014-2015 agree closely with Hubble Space Telescope (HST) measurements made during the same epoch Bedin et al. 2017, and the derived mutual orbit agrees with those measurements to within the HST uncertainties of $0.3 - 0.4$ milliarcseconds.
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