We carried out targeted ALMA observations of 129 fields in the COSMOS region at 1.25mm, detecting 152 galaxies at S/N>=5 with an average continuum RMS of 150 {mu}Jy. These fields represent a S/N-limited sample of AzTEC/ASTE sources with 1.1mm S/N>=4 over an area of 0.72 square degrees. Given ALMA's fine resolution and the exceptional spectroscopic and multiwavelength photometric data available in COSMOS, this survey allows us unprecedented power in identifying submillimeter galaxy counterparts and determining their redshifts through spectroscopic or photometric means. In addition to 30 sources with prior spectroscopic redshifts, we identified redshifts for 113 galaxies through photometric methods and an additional nine sources with lower limits, which allowed a statistically robust determination of the redshift distribution. We have resolved 33 AzTEC sources into multi-component systems and our redshifts suggest that nine are likely to be physically associated. Our overall redshift distribution peaks at z~2.0 with a high-redshift tail skewing the median redshift to z^~^=2.48+/-0.05. We find that brighter millimeter sources are preferentially found at higher redshifts. Our faintest sources, with S_1.25mm_1.8mJy, have a median redshift of z^~^=3.08+/-0.17. After accounting for spectral energy distribution shape and selection effects, these results are consistent with several previous submillimeter galaxy surveys, and moreover, support the conclusion that the submillimeter galaxy redshift distribution is sensitive to survey depth.