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In this paper we present the CMZoom Survey's catalog of compact sources (< 10'', ~0.4pc) within the Central Molecular Zone (CMZ). CMZoom is a Submillimeter Array (SMA) large program designed to provide a complete and unbiased map of all high column density gas (N(H$_2$) $\geq$ 10$^{23}$ cm$^{-2}$) of the innermost 500pc of the Galaxy in the 1.3mm dust continuum. We generate both a robust catalog designed to reduce spurious source detections, and a second catalog with higher completeness, both generated using a pruned dendrogram. In the robust catalog, we report 285 compact sources, or 816 in the high completeness catalog. These sources have effective radii between 0.04-0.4 pc, and are the potential progenitors of star clusters. The masses for both catalogs are dominated by the Sagittarius B2 cloud complex, where masses are likely unreliable due to free-free contamination, uncertain dust temperatures, and line-of-sight confusion. Given the survey selection and completeness, we predict that our robust catalog accounts for more than ~99% of compact substructure capable of forming high mass stars in the CMZ. This catalog provides a crucial foundation for future studies of high-mass star formation in the Milky Way's Galactic Center.