Spin-oriented, normally emitted Cu 3d photoelectrons, generated by circularly polarized synchrotron radiation in the energy range of 10-24 eV, are transmitted through perpendicularly magnetized ultrathin fcc iron films on Cu(100). Spin-dependent scattering of electrons of majority and minority spin with respect to the Fe magnetization direction results in an intensity asymmetry, which is 22% for electrons with kinetic energies of 5 eV and an Fe film thickness of 4.3 monolayers. Exploiting this effect, the iron film is used as a very efficient spin detector to investigate spin-dependently the relativistic bulk band structure of the Cu 3d states in Fe/Cu(100). Good agreement with fully relativistic band-structure calculations is found.