We present the 250, 350, and 500 Delta *mm detection of bright submillimeter emission in the direction of the Bullet Cluster measured by the Balloon-borne Large-Aperture Submillimeter Telescope (BLAST). The 500 Delta *mm centroid is coincident with an AzTEC 1.1 mm point-source detection at a position close to the peak lensing magnification produced by the cluster. However, the 250 Delta *mm and 350 Delta *mm centroids are elongated and shifted toward the south with a differential shift between bands that cannot be explained by pointing uncertainties. We therefore conclude that the BLAST detection is likely contaminated by emission from foreground galaxies associated with the Bullet Cluster. The submillimeter redshift estimate based on 250-1100 Delta *mm photometry at the position of the AzTEC source is z phot = 2.9+0.6 -0.3, consistent with the infrared color redshift estimation of the most likely Infrared Array Camera counterpart. These flux densities indicate an apparent far-infrared (FIR) luminosity of L FIR = 2 X 1013 L. When the amplification due to the gravitational lensing of the cluster is removed, the intrinsic FIR luminosity of the source is found to be L FIR <= 1012 L, consistent with typical luminous infrared galaxies.