The effect of a strong magnetic field on neutron stars or white dwarfs is calculated for Thomson scattering in a strong magnetic field with radiation damping. The Stokes's parameters for the scattered radiation are computed explicitly in terms of the state of polarization of the incident wave, the electron-cyclotron frequency, the angle of incidence, and the angle of scattering. The criterion for the magnetic field to substantially affect the Stokes's parameters is that the photon frequency be less than the electron-cyclotron frequency. The effects of classical radiation damping are explicitly taken into account by using the Abraham-Lorentz equation. The corresponding quantum mechanical treatment for Comptom scattering in a relativistic electron gas is now under investigation.