  We present ALMA 850 $\mu$m continuum observations of the Orion Nebula Cluster
that provide the highest angular resolution ($\sim 0\rlap{.}''1 \approx 40$ AU)
and deepest sensitivity ($\sim 0.1$ mJy) of the region to date. We mosaicked a
field containing $\sim 225$ optical or near-IR-identified young stars, $\sim
60$ of which are also optically-identified "proplyds". We detect continuum
emission at 850 $\mu$m towards $\sim 80$% of the proplyd sample, and $\sim 50$%
of the larger sample of previously-identified cluster members. Detected objects
have fluxes of $\sim 0.5$-80 mJy. We remove sub-mm flux due to free-free
emission in some objects, leaving a sample of sources detected in dust
emission. Under standard assumptions of isothermal, optically thin disks,
sub-mm fluxes correspond to dust masses of $\sim 0.5$ to 80 Earth masses. We
measure the distribution of disk sizes, and find that disks in this region are
particularly compact. Such compact disks are likely to be significantly
optically thick. The distributions of sub-mm flux and inferred disk size
indicate smaller, lower-flux disks than in lower-density star-forming regions
of similar age. Measured disk flux is correlated weakly with stellar mass,
contrary to studies in other star forming regions that found steeper
correlations. We find a correlation between disk flux and distance from the
massive star $\theta^1$ Ori C, suggesting that disk properties in this region
are influenced strongly by the rich cluster environment.
