We demonstrate that it is possible to tune the Rashba energy, introduced by a strong spin-orbit splitting, and the Fermi energy in a two-dimensional electron gas by a controlled change of stoichiometry in an artificial surface alloy. In the BixPb1-x/Ag(111) surface alloy, the spin-orbit interaction maintains a dramatic influence on the band dispersion for arbitrary Bi concentration x, as is shown by angle-resolved photoelectron spectroscopy. The Rashba energy E-R and the Fermi energy E-F can be tuned to achieve values larger than one for the ratio E-R/E-F, which opens up the possibility for observing phenomena, such as corrections to the Fermi liquid or a superconducting state. Relativistic first-principles calculations explain the experimental findings.