  The Gaia space telescope allows for unprecedented accuracy for astrometric
measurements of stars in the Galaxy. In this work, we explore the sensitivity
of Gaia to detect primordial black hole (PBH) dark matter through the
distortions that PBHs would create in the apparent trajectories of background
stars, an effect known as astrometric microlensing (AML). We present a novel
calculation of the lensing probability, and we combine this with the existing
publicly released Gaia eDR3 stellar catalog to predict the expected rate of AML
events that Gaia will see. We also compute the expected distribution of a few
event observables, which will be useful for reducing backgrounds. Assuming that
the astrophysical background rate of AML like events due to other sources is
negligible, we then compute the potential exclusion that could be set on the
parameter space of PBHs with a monochromatic mass function. We find that Gaia
is sensitive to PBHs in the range of $0.4~M_\odot$ - $5\times10^7~M_\odot$, and
has peak sensitivity to PBHs of $\sim 10~M_\odot$ for which it can rule out as
little as a fraction $3\times10^{-4}$ of dark matter composed of PBHs. With
this exquisite sensitivity, Gaia has the potential to rule out a PBH origin for
the gravitational wave signals seen at LIGO/Virgo. Our novel calculation of the
lensing probability includes for the first time, the effect of intermediate
duration lensing events, where the lensing event lasts for a few years, but for
a period which is still shorter than the Gaia mission lifetime. The lower end
of our predicted mass exclusion is especially sensitive to this class of
events. As and when time-series data for Gaia is released, and once we have a
better understanding of the backgrounds, our prediction of the lensing rate and
event observable distributions will be useful to estimate the true
exclusion/discovery of the PBH parameter space utilizing this data.
