Using a recently developed unified theory of coherence and polarization [ E. Wolf, Phys. Lett., A312, 263 - 267 ( 2003)] it is shown that a spatially completely coherent random electromagnetic beam can have a degree of polarization P which may take on any value ( 0 less than or equal to P less than or equal to 1) at points in the beam. In particular the beam can be fully polarized at one point and completely unpolarized or partially polarized at another point. This result also holds for electromagnetic beams which are not completely spatially coherent.