The MACHO project carries out regular photometric monitoring of millions of stars in the Magellanic Clouds and Galactic Bulge, to search for very rare gravitational microlensing events due to compact objects in the galactic halo and disk. A preliminary analysis of one field in the Galactic Bulge, containing {$sim430,000$} stars observed for 190 days, reveals four stars which show clear evidence for brightenings which are time-symmetric, achromatic in our two passbands, and have shapes consistent with gravitational microlensing. This is significantly higher than the $sim 1$ event expected from microlensing by known stars in the disk. If all four events are due to microlensing, a 95% confidence lower limit on the optical depth towards our bulge field is $1.3 times 10^{-6}$, and a ``best fit value is $tau approx 1.6 times 10^{-6}/epsilon$,where $epsilon$ is the detection efficiency of the experiment, and $epsilon < 0.4$. If the true optical depth is close to the ``best fit value, possible explanations include a ``maximal disk which accounts for most of the galactic circular velocity at the solar radius, a halo which is centrally concentrated, or bulge-bulge microlensing.
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