Deep 1.1mm continuum observations of 1E0657-56 (the `Bullet Cluster') taken with the millimeter-wavelength camera AzTEC on the 10-m Atacama Submillimeter Telescope Experiment (ASTE), have revealed an extremely bright (S1.1mm = 15.9mJy) unresolved source. This source, MMJ065837-5557.0, lies close to a maximum in the density of underlying mass distribution, towards the larger of the two interacting clusters as traced by the weak-lensing analysis of Clowe et al. Using optical-infrared (IR) colours, we argue that MMJ065837-5557.0 lies at a redshift of z = 2.7 +/- 0.2. A lensing-derived mass model for the Bullet Cluster shows a critical line (caustic) of magnification within a few arcsec of the AzTEC source, sufficient to amplify the intrinsic millimetre-wavelength flux of the AzTEC galaxy by a factor of >>20. After subtraction of the foreground cluster emission at 1.1mm due to the Sunyaev-Zel'dovich effect, and correcting for the magnification, the rest-frame far-IR luminosity of MMJ065837-5557.0 is <=1012Lsolar, characteristic of a luminous infrared galaxy (LIRG). We explore various scenarios to explain the colours, morphologies and positional offsets between the potential optical and IR counterparts, and their relationship with MMJ065837-5557.0. Until higher resolution and more sensitive (sub)millimetre observations are available, the detection of background galaxies close to the caustics of massive lensing clusters offers the only opportunity to study this intrinsically faint millimetre-galaxy population.