  In astrometric binaries, the presence of a dark, unseen star can be inferred
from the gravitational pull it induces on its luminous binary companion. While
the orbit of such binaries can be characterized with precise astrometric
measurements, constraints made from astrometry alone are not enough to measure
the component masses. In this work, we determine the precision with which Gaia
can astrometrically measure the orbits and -- with additional observations --
the component masses, for luminous stars hosting hidden companions. Using
realistic mock Gaia observations, we find that Gaia can precisely measure the
orbits of binaries hosting hidden brown-dwarfs out to tens of pc and hidden
white dwarf and neutron star companions at distances as far as several hundred
pc. Heavier black hole companions may be measured out to 1 kpc or farther. We
further determine how orbital period affects this precision, finding that Gaia
can characterize orbits with periods as short as 10 days and as long as a few
10$^3$ days, with the best measured orbits having periods just short of Gaia's
mission lifetime. Extending Gaia's nominal five-year mission lifetime by an
additional five years not only allows for the measurement of longer period
orbits, but those longer period binaries can be seen at even greater distances.
