The Chern-Simons Lagrangian has been studied previously in (2+1)-dimensional spacetime, where it is both gauge and Lorentz invariant. In 3+1 dimensions, this term couples the dual electromagnetic tensor to an external four-vector. If we take this four-vector to be fixed, the term is gauge invariant but not Lorentz invariant. In this paper, we examine both the theoretical consequences of such a modification and observational limits we can put on its magnitude. The Chern-Simons term would rotate the plane of polarization of radiation from distant galaxies, an effect which is not observed. From the observations we deduce that the magnitude of the vector is <1.7×10-42h0 GeV, where h0 is the Hubble constant in units of 100 km sec-1 Mpc-1.