The electronic structure of ultrathin Ag(111) films covered with a square root(3) x square root(3)-Bi/Ag ordered alloy was investigated by means of spin- and angle-resolved photoemission spectroscopy. Surface-state (SS) bands, spin split by the Rashba interaction, selectively couple to the quantum-well state (QWS) bands, originally spin degenerate, in the metal film. Gaps are found to open between QWS and SS with parallel spins, while free-electron-like QWS dispersions are observed for antiparallel spin configurations. The present results demonstrate that in a nonmagnetic metal film the spin degeneracy of the valence levels can be lifted by hybridization with Rashba-type SS bands.