The Fermi and Rashba energies of surface states in the Bi_xPb_{1-x}/Ag(111) alloy can be tuned simultaneously by changing the composition parameter x. We report on unconventional Fermi surface spin textures observed by spin and angle-resolved photoem
ission spectroscopy {that are correlated with a topological transition of the Fermi surface occurring at x=0.5. We show that the surface states remain fully spin polarized upon alloying and that the spin polarization vectors are approximately tangential to the constant energy contours. We discuss the implications of the topological transition for the transport of spin.
