We present the first results from a 1.1mm 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 270arcmin^2^ field to a 1{sigma} depth of 0.48-0.73mJy/beam, 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 (1{sigma}~1mJy/beam) outer region of the map.