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A simplified experimental technique, utilizing Compton scattering as the analyzing process, has been developed for the measurement of the plane polarization of gamma radiation emitted from oriented nuclei. Nuclei of ${\mathrm{Ce}}^{139}$ and ${\mathrm{Ce}}^{141}$ were aligned in a neodymium ethylsulfate lattice at low temperatures by the magnetic-hyperfine-structure method. The plane polarization of the emitted gamma radiation was measured as a function of temperature. The measured polarization of this radiation was found to be consistent with published results.