We report evidence for a phase transition between a spin-unpolarized state and a spin-polarized state in a two-dimensional electron gas in tilted magnetic fields. This effect is revealed in the magnetoresistance at mK temperatures, where the coincidence of the spin-resolved Landau levels N = 1 up and N = 0 down completely vanishes in samples of sufficiently high mobility, as predicted in recent theoretical studies. The effect is ascribed to electron-electron interactions.