(Abridged) We compare recent results from X-ray, strong lensing, weak lensing, and optical observations with numerical simulations of the merging galaxy cluster 1E0657-56. X-ray observations reveal a bullet-like subcluster with a prominent bow shock, while lensing results show that the positions of the total mass peaks are consistent with the centroids of the collisionless galaxies (and inconsistent with the X-ray brightness peaks). Previous studies, based on older observational datasets, have placed upper limits on the self-interaction cross-section of dark matter per unit mass, sigma/m, using simplified analytic techniques. In this work, we take advantage of new, higher-quality observational datasets by running N-body simulations of 1E0657-56 that include the effects of self-interacting dark matter, and comparing the results with observations. Furthermore, the recent data allow for a new independent method of constraining sigma/m, based on the non-observation of an offset between the bullet subcluster mass peak and galaxy centroid. This new method places an upper limit (68% confidence) of sigma/m < 1.25 cm^2/g. If we make the assumption that the subcluster and the main cluster had equal mass-to-light ratios prior to the merger, we derive our most stringent constraint of sigma/m < 0.7 cm^2/g, which comes from the consistency of the subcluster's observed mass-to-light ratio with the main cluster's, and with the universal cluster value, ruling out the possibility of a large fraction of dark matter particles being scattered away due to collisions. Our limit is a slight improvement over the previous result from analytic estimates, and rules out most of the 0.5 - 5cm^2/g range invoked to explain inconsistencies between the standard collisionless cold dark matter model and observations.