We measured the dispersion of an unoccupied surface state on Pt(111) by imaging scattering states at point defects and step edges using scanning tunneling spectroscopy. By comparison to first-principles electronic structure calculations the state is assigned to an sp-derived surface band at the lower edge of the projected bulk band gap. In dI/dV(V) curves, the onset of the surface-state band appears as a rather broad feature. Its shape results from two spin-orbit split branches with nearly linear dispersion, one of them merging into bulk states at higher energies.