The polarization of the 100 micron continuum emission has been measured at 14 positions in the dense, warm molecular cloud associated with the arched filaments, or the 'bridge', of the radio arc near the Galactic center. At all positions the percent polarization is found to be quite large, ranging up to 6.5 percent. The polarization is interpreted in terms of thermal emission by magnetically aligned dust grains. The directions of the polarization vectors then indicate that the magnetic field is (1) parallel to the long dimension of the thermal radio filaments, and (2) very uniform on scales of 1-10 pc. Of several explanations for the inferred field geometry, the simplest is that it results from the unusually large dynamical shear in the emitting cloud.