  [Abridged] We present the first results from a 1.1 mm confusion-limited map
of the GOODS-S field taken with AzTEC on the ASTE telescope. We imaged a 270
sq. arcmin field to a 1\sigma depth of 0.48 - 0.73 mJy/beam, making this one of
the deepest blank-field surveys at mm-wavelengths ever achieved. Although our
GOODS-S map is extremely confused, we demonstrate that our source
identification and number counts analyses are robust, and the techniques
discussed in this paper are relevant for other deeply confused surveys. We find
a total of 41 dusty starburst galaxies with S/N >= 3.5 within this uniformly
covered region, where only two are expected to be false detections. We derive
the 1.1mm number counts from this field using both a "P(d)" analysis and a
semi-Bayesian technique, and find that both methods give consistent results.
Our data are well-fit by a Schechter function model with (S', N(3mJy), \alpha)
= (1.30+0.19 mJy, 160+27 (mJy/deg^2)^(-1), -2.0). Given the depth of this
survey, we put the first tight constraints on the 1.1 mm number counts at
S(1.1mm) = 0.5 mJy, and we find evidence that the faint-end of the number
counts at S(850\mu m) < 2.0 mJy from various SCUBA surveys towards lensing
clusters are biased high. In contrast to the 870 \mu m survey of this field
with the LABOCA camera, we find no apparent under-density of sources compared
to previous surveys at 1.1 mm. Additionally, we find a significant number of
SMGs not identified in the LABOCA catalogue. We find that in contrast to
observations at wavelengths < 500 \mu m, MIPS 24 \mu m sources do not resolve
the total energy density in the cosmic infrared background at 1.1 mm,
demonstrating that a population of z > 3 dust-obscured galaxies that are
unaccounted for at these shorter wavelengths potentially contribute to a large
fraction (~2/3) of the infrared background at 1.1 mm.
