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 ~3x deeper and with a beam area ~200x 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 ~1".6 resolution allows us to measure a size of ~9 kpc x 5 kpc for the rest-frame ~300 mu m emission region in one resolved SMG, implying a star formation rate surface density of 80 M sub(middot in circle) yr super(-1) kpc super(-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.