We present the counts of luminous submillimeter (sub-mm) galaxies from an analysis of our completed survey of the distant universe seen through lensing clusters. This survey uses massive cluster lenses with well-constrained mass models to obtain a magnified view of the background sky. This both increases the sensitivity of our sub-mm maps and reduces the effects of source confusion. Accurate lens models are used to correct the observed sub-mm source counts for the lens amplification. We show that the uncertainties associated with this correction do not dominate the final errors. We present sub-mm counts derived from two independent methods: a direct inversion of the observed sources, which are corrected individually for lens amplification, and a Monte Carlo simulation of our observations using a parametric model for the background counts, which is folded through the lens models and incompleteness estimates to determine best-fitting values of the count parameters. Both methods agree well and confirm the robustness of our analysis. Detections that are identified with galaxies in the lensing clusters in deep optical images are removed prior to our analysis, and the results are insensitive both to the details of the correction and to the redshift distribution of the detections. We present the 850 µm counts at flux densities between 0.5 and 8 mJy. The count of galaxies brighter than 4 mJy is 1500±700 deg–2, in agreement with the value of 2500±1400 deg–2 reported by Smail, Ivison, & Blain in 1997. The most accurate 850 µm count is determined at 1 mJy: 7900±3000 deg–2. All quoted errors include both Poisson and systematic terms. These are the deepest sub-mm counts published, and they are not subject to source confusion because the detected galaxies are separated and magnified by the lens. Down to the 0.5 mJy limit of our counts, the resolved 850 µm background radiation intensity is (5±2)x10–10 W.m–2.sr–1, comparable to the current COBE estimate of the background. This indicates that the bulk of the 850 µm background radiation originated in distant ultraluminous galaxies.