We present an Atacama Large Millimeter/submillimeter Array ( ALMA ) Cycle 0 survey of 126 submillimeter sources from the LABOCA ECDFS Submillimeter Survey ( LESS ) . Our 870 \mu m survey with ALMA ( ALESS ) has produced maps \sim 3 \times deeper and with a beam area \sim 200 \times 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 discuss the creation of the ALESS submillimeter galaxy ( SMG ) catalog , including the main sample of 99 SMGs and a supplementary sample of 32 SMGs . We find that at least 35 % ( possibly up to 50 % ) of the detected LABOCA sources have been resolved into multiple SMGs , and that the average number of SMGs per LESS source increases with LESS flux density . Using the ( now precisely known ) SMG positions , we empirically test the theoretical expectation for the uncertainty in the single–dish source positions . We also compare our catalog to the previously predicted radio/mid–infrared counterparts , finding that 45 % of the ALESS SMGs were missed by this method . Our \sim 1.6 ^ { \prime \prime } resolution allows us to measure a size of \sim 9 kpc \times 5 kpc for the rest–frame \sim 300 \mu m emission region in one resolved SMG , implying a star formation rate surface density of 80 M _ { \odot } yr ^ { -1 } kpc ^ { -2 } , and we constrain the emission regions in the remaining SMGs to be < 10 kpc . As the first statistically reliable survey of SMGs , this will provide the basis for an unbiased multiwavelength study of SMG properties . Key words : galaxies : starburst – galaxies : high-redshift – submillimeter – catalogs