Near-infrared spectroscopic surveys have uncovered a population of short-period, blended-light spectral binaries composed of low-mass stars and brown dwarfs. These systems are amenable to orbit determination and individual mass measurements via astrometric monitoring. Here, we present first results of a multiyear campaign to obtain high-precision absolute astrometry for spectral binaries using the Gemini-South and Gemini-North GMOS imagers. We measure the complete astrometric orbits for two systems: 2M0805+48 and 2M1059-21. Our astrometric orbit of 2M0805+48 is consistent with its 2-yr radial velocity orbit determined previously and we find a mass of $66^{+5}_{-14} M_\mathrm{Jup}$ for its T5.5 companion. For 2M1059-21, we find a 1.9-yr orbital period and a mass of $67^{+4}_{-5} M_\mathrm{Jup}$ for its T3.5 companion. We demonstrate that sub-milliarcsecond absolute astrometry can be obtained with both GMOS imagers and that this is an efficient avenue for confirming and characterizing ultracool binary systems.