We report the first results of AS2UDS, an 870 μm continuum survey with the Atacama Large Millimeter/Submillimeter Array (ALMA) of a total area of ˜50 arcmin2 comprising a complete sample of 716 submillimeter sources drawn from the SCUBA-2 Cosmology Legacy Survey (S2CLS) map of the UKIDSS/UDS field. The S2CLS parent sample covers a 0.96 degree2 field at σ 850 = 0.90 ± 0.05 mJy beam-1. Our deep, high-resolution ALMA observations with σ 870 ˜ 0.25 mJy and a 0.″15-0.″30 FWHM synthesized beam, provide precise locations for 695 submillimeter galaxies (SMGs) responsible for the submillimeter emission corresponding to 606 sources in the low-resolution, single-dish map. We measure the number counts of SMGs brighter than S 870 ≥ 4 mJy, free from the effects of blending and show that the normalization of the counts falls by 28% ± 2% in comparison with the SCUBA-2 published counts, but that the shape remains unchanged. We determine that {44}-14+16% of the brighter single-dish sources with S 850 ≥ 9 mJy consist of a blend of two or more ALMA-detectable SMGs brighter than S 870 ˜ 1 mJy (corresponding to a galaxy with a total-infrared luminosity of L IR ≳ 1012 L ⊙), in comparison with 28% ± 2% for the single-dish sources at S 850 ≥ 5 mJy. Using the 46 single-dish submillimeter sources that contain two or more ALMA-detected SMGs with photometric redshifts, we show that there is a significant statistical excess of pairs of SMGs with similar redshifts (