The first experimental realization of a photon Bose-Einstein condensate was demonstrated more than a decade ago. However, the polarization of the condensate has not been fully understood and measured in this weakly driven-dissipative system. In this article, we experimentally investigate the polarization of thermal and condensed light depending on the power and polarization of the pump beam. With full control over the polarization of the pump light, it is possible to create arbitrary states on the surface of the Poincaré sphere. We show that, in agreement with previous theoretical work, there is a remarkable increase in the polarization strength of the condensate above the threshold for a fully linearly polarized pump light, indicating that the weak drive lifts the polarization degeneracy of the system. Above a certain threshold also the degenerate orthogonal polarized state of the cavity is occupied, limiting the degree of polarization to approximately 90%.