Based on our initial low-resolution observations from the LESS survey we have awarded ALMA observing time in Cycle-0 to produce the first complete high-resolution follow-up catalog of 126 SMGs in LESS. Our 870μm survey with ALMA has produced maps ∼3× deeper and with a beam area ∼200× smaller than the original LESS observations, doubling the current number of interferometrically–observed submillimeter sources. The high resolution of these maps allows us to resolve sources that were previously blended and accurately identify the origin of the submillimeter emission. We show that our ALMA-derived SMG counts broadly agree with the submillimeter source counts from previous, lower-resolution single-dish surveys, demonstrating that the bulk of the submillimeter sources are not caused by blending of unresolved SMGs. The difficulty that well-constrained theoretical models have in reproducing the high- surface densities of SMGs, thus remains. However, our observations do show that all of the very brightest sources in the LESS sample, S870μm > 12 mJy, comprise emission from multiple, fainter SMGs, each with 870μm fluxes of <∼ 9 mJy.