The technique and results of a measurement of the linear polarization of the cosmic background radiation at a wavelength of 9 mm are discussed. Data taken between 1978 May and 1980 February from both the Northern Hemisphere (Berkeley latitude 38 deg N) and the Southern Hemisphere (Lima latitude 12 deg S) over 11 declinations from -37 to +63 deg show the radiation to be essentially unpolarized over all areas surveyed. Fitting all data gives the 95% confidence level limit on a linearly polarized component of 0.3 mK for spherical harmonics through third order. A fit of all data to the anisotropic axisymmetric model of Rees (1968) yields a 95% confidence level limit of 0.15 mK for the magnitude of the polarized component. Constraints on various cosmological models are discussed in light of these limits.