The spatial distribution of the polarized component of the thermal dust emission from the core of the Orion molecular cloud has been mapped over 15 positions at a wavelength of 1300-mu-m using the MILLIPOL instrument on the NRAO 12 m telescope. The chief findings of this study are three-fold. First, the polarization directions deduced for the map positions are quite similar, implying that the embedded magnetic field is strong (of the order of 1 mG) and uniform in direction over the core of the cloud. The angular dispersion, corrected for measurement uncertainty, is approximately 9-degrees, roughly identical to that seen in optical starlight polarization studies of dark clouds whose evolution is believed to have been dominated by magnetic fields. Second, the mean polarization percentage, 4.6%, is quite high, and two positions show polarization percentages of nearly 8%, the highest yet detected for polarized thermal dust emission. Finally, the polarization of the BN/KL position (2.70% +/- 0.25%) was found to be significantly lower than the polarizations toward positions 30" outside BN/KL.