The (114) surface of the semimetal Bi is found to support a quasi-one-dimensional, metallic surface state. As required by symmetry, the state is degenerate along the Gamma-Y line of the surface Brillouin zone with a highest binding energy of approximately 150 meV. In the Gamma-X direction the degeneracy is lifted by the strong spin-orbit splitting in Bi, as directly shown by spin-resolved photoemission. This results in a Fermi contour consisting of two closely separated, parallel lines of opposite spin direction. It is argued that similar states on related insulators would give rise to a one-dimensional quantum spin Hall effect.