We observe a giant spin-orbit splitting in the bulk and surface states of the noncentrosymmetric semiconductor BiTeI. We show that the Fermi level can be placed in the valence or in the conduction band by controlling the surface termination. In both cases, it intersects spin-polarized bands, in the corresponding surface depletion and accumulation layers. The momentum splitting of these bands is not affected by adsorbate-induced changes in the surface potential. These findings demonstrate that two properties crucial for enabling semiconductor-based spin electronics-a large, robust spin splitting and ambipolar conduction-are present in this material.