We present the first results from a confusion-limited map of the Great Observatories Origins Deep Survey-South (GOODS-S) taken with the AzTEC camera on the Atacama Submillimeter Telescope Experiment. We imaged a field to a 1σ depth of 0.48-0.73 mJy beam-1, making this one of the deepest blank-field surveys at mm-wavelengths ever achieved. Although by traditional standards our GOODS-S map is extremely confused due to a sea of faint underlying sources, we demonstrate through simulations 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 signal-to-noise ratios S/N ≥ 3. 5 within this uniformly covered region, where only two are expected to be false detections, and an additional seven robust source candidates located in the noisier (1σ ≈ 1 mJy beam-1) outer region of the map. We derive the 1.1 mm number counts from this field using two different methods: a fluctuation or "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. Given the depth of this survey, we put the first tight constraints on the 1.1 mm number counts at S1.1 mm = 0.5 mJy, and we find evidence that the faint end of the number counts at from various SCUBA surveys towards lensing clusters are biased high. In contrast to the 870 μm survey of this field with the LABOCA camera, we find no apparent underdensity of sources compared to previous surveys at 1.1 mm; the estimates of the number counts of SMGs at flux densities >1 mJy determined here are consistent with those measured from the AzTEC/SHADES survey. Additionally, we find a significant number of SMGs not identified in the LABOCA catalogue. We find that in contrast to observations at λ ≤ 500 μm, MIPS 24 μm 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. © 2010 The Authors. Journal compilation © 2010 RAS.