The electrical properties of nanostructures are extremely sensitive to their surface condition. In very thin two-dimensional crystalline-semiconductor sheets, termed nanomembranes, the influence of the bulk is diminished, and the electrical conductance becomes exquisitely responsive to the structure of the surface and the type and density of defects there. Its understanding therefore requires a precise knowledge of the surface condition. Here we report measurements, using nanomembranes, that demonstrate direct charge transport through the p* band of the clean reconstructed Si(001) surface. We determine the charge carrier mobility in this band. These measurements, performed in ultra-high vacuum to create a truly clean surface, lay the foundation for a quantitative understanding of the role of extended or localized surface states, created by surface structure, defects or adsorbed atoms/molecules, in modifying charge transport through semiconductor nanostructures.