For radiative electron capture into the K shell of bare uranium ions, a study of the polarization properties has been performed. For this purpose a position sensitive germanium detector has been used as an efficient Compton polarimeter. This enabled us to measure the degree of linear polarization by analyzing Compton scattering inside the detector and to determine the orientation of the polarization plane. Depending on the observation angle and the beam energy used, the radiation is found to be linearly polarized by up to 80%. In all cases studied, the plane of polarization coincides with the collision plane. The results will be discussed in the context of rigorous relativistic calculations, showing that relativistic effects tend to lead to a depolarization of the radiation emitted.