  We show that a small, but \textit{measurable} shift in the eclipse mid-point
time of eclipsing binary (EBs) stars of $\sim$ 0.1 seconds over a decade
baseline can be used to directly measure the Galactic acceleration of stars in
the Milky Way at $\sim$ kpc distances from the Sun. We consider contributions
to the period drift rate from dynamical mechanisms other than the Galaxy's
gravitational field, and show that the Galactic acceleration can be reliably
measured using a sample of $\textit{Kepler}$ EBs with orbital and stellar
parameters from the literature. Given the uncertainties on the formulation of
tidal decay, our approach here is necessarily approximate, and the contribution
from tidal decay is an upper limit assuming the stars are not tidally
synchronized. We also use simple analytic relations to search for well-timed
sources in the \textit{Kepler} field, and find $\sim$ 70 additional detached
EBs with low eccentricities that have estimated timing precision better than 1
second. We illustrate the method with a prototypical, precisely timed EB using
an archival \textit{Kepler} light curve and a modern synthetic \textit{HST}
light curve (which provides a decade baseline). This novel method establishes a
realistic possibility for obtaining fundamental Galactic parameters using
eclipse timing to measure Galactic accelerations, along with other emerging new
methods, including pulsar timing and extreme precision radial velocity
observations. This acceleration signal grows quadratically with time.
Therefore, given baselines established in the near-future for distant EBs, we
can expect to measure the period drift in the future with space missions like
\textit{JWST} and the \textit{Roman Space Telescope}.
