Linearly polarized gamma rays were produced with the aid of a special betatron doughnut which made it possible to limit the effective thickness of the bremsstrahlung-producing Al and Pt targets. The linear polarization of 15.1-Mev bremsstrahlung gamma rays was detected by measuring the azimuthal angular distribution of gamma rays which had been scattered elastically from the well-known nuclear level in C12. For the particular effective thickness of Al that was used, the maximum polarization of 15.1-Mev gamma rays in the bremsstrahlung of 25-Mev electrons was 1.53+/-0.05. (This corresponds to 21% in the more conventional polarization notation.) Both this observed maximum (which occurred at an angle of 1.4°) and the polarization measured at four other angles confirm theoretical predictions. With the platinum target that was used, the observed polarization of 1.28+/-0.04 was about 10% lower than the value predicted by theoretical calculations which use the Born approximation and neglect screening. However, available screening and Coulomb corrections bring the theoretical predictions into essential agreement with the experimental values. Since the elastic scattering pattern of gamma rays shows unambiguously that the strong photon scattering level at 15.1 Mev in C12 is excited by magnetic dipole radiation, this level has spin 1 and even parity.