We determine the quantum phase diagram of the one-dimensional Hubbard model with bond-charge interaction X in addition to the usual Coulomb repulsion U> 0 at half-filling. For large enough X< t the model shows three phases. For large U the system is in the spin-density wave phase as in the usual Hubbard model. As U decreases, there is first a spin transition to a spontaneously dimerized bond-ordered wave phase and then a charge transition to a novel phase in which the dominant correlations at large distances correspond to an incommensurate singlet superconductor.