Spin-orbit coupling can give rise to spin-split electronic states without a ferromagnet or an external magnetic field. We create large spin-orbit splittings in a Au and Ag monolayer on W(110) and show that the size of the splitting does not depend on the atomic number of the Au or Ag overlayer but of the W substrate. Spin- and angle-resolved photoemission and Fermi-surface scans reveal that the overlayer states acquire spin polarization through spin-dependent overlayer-substrate hybridization.