  We carried out targeted ALMA observations of 129 fields in the COSMOS region
at 1.25 mm, detecting 152 galaxies at S/N$\geq$5 with an average continuum RMS
of 150 $\mu$Jy. These fields represent a S/N-limited sample of AzTEC / ASTE
sources with 1.1 mm S/N$\geq$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\sim$2.0 with a high redshift tail
skewing the median redshift to $\tilde{z}$=2.48$\pm$0.05. We find that brighter
millimeter sources are preferentially found at higher redshifts. Our faintest
sources, with S$_{1.25 \rm mm}$<1.25 mJy, have a median redshift of
$\tilde{z}$=2.18$\pm$0.09, while the brightest sources, S$_{1.25 \rm mm}$>1.8
mJy, have a median redshift of $\tilde{z}$=3.08$\pm$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.
