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The Doppler spectrum of electromagnetic radiation scattered by a turbulent plasma is calculated for the case in which the incident wave is a high frequency microwave or a laser beam and is not of sufficient intensity to heat the plasma. The differential scattering cross section is expressed in terms of the power spectrum of the current density fluctuations, and the wave equation and the equations of continuity and momentum for the electrons are analyzed by means of the repeated cascade theory of turbulence. The equation governing the current density spectrum in the universal equilibrium range consists of three terms: a term representing the excitation of the spectrum near the wave number of the incident wave, a term which transfers energy across the spectrum at a rate controlled by an eddy viscosity, and a molecular dissipation term. Solutions for the spectrum are obtained in the inertial and dissipation subranges which reduce to Kolmogorov and Heisenberg type spectra, respectively, in the limits of zero scattering or for wave numbers far from the wave number of the incident wave. Near the frequency ω0 of the incident wave, the differential cross sections, or equivalently Doppler spectra, for scattering at frequency ω vary exponentially as − Δω−5/3 in the inertial subrange and as −Δω3 in the dissipation subrange, where Δω = ω - ω0. The characteristic frequencies controlling the Doppler half widths depend upon the scattering angle, the intensity of the incident wave, and the root-mean-square electron density fluctuation.