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(abridged) We determine the disk mass distribution around 336 stars in the young Orion Nebula cluster by imaging a 2.5' x 2.5' region in 3 mm continuum emission with the Owens Valley Millimeter Array. For this sample of 336 stars, we observe 3 mm emission above the 3-sigma noise level toward ten sources, six of which have also been detected optically in silhouette against the bright nebular background. In addition, we detect 20 objects that do not correspond to known near-IR cluster members. Comparisons of our measured fluxes with longer wavelength observations enable rough separation of dust emission from thermal free-free emission, and we find substantial dust emission toward most objects. For the ten objects detected at both 3 mm and near-IR wavelengths, eight exhibit substantial dust emission. Excluding the high-mass stars and assuming a gas-to-dust ratio of 100, we estimate circumstellar masses ranging from 0.13 to 0.39 Msun. For the cluster members not detected at 3 mm, images of individual objects are stacked to constrain the mean 3 mm flux of the ensemble. The average flux is detected at the 3-sigma confidence level, and implies an average disk mass of 0.005 Msun, comparable to the minimum mass solar nebula. The percentage of stars in Orion surrounded by disks more massive than ~0.1 Msun is consistent with the disk mass distribution in Taurus, and we argue that massive disks in Orion do not appear to be truncated through close encounters with high-mass stars. Comparison of the average disk mass and number of massive dusty structures in Orion with similar surveys of the NGC 2024 and IC 348 clusters constrains the evolutionary timescales of massive circumstellar disks in clustered environments.