Using quantum field theory and bosonization, 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 at half-filling, for small values of the interactions. We show that it is essential to take into account formally irrelevant terms of order X. They generate relevant terms proportional to X2 in the flow of the renormalization group (RG). These terms are calculated using operator product expansions. The model shows three phases separated by a charge transition at U=Uc and a spin transition at U=Us>Uc. For U<Uc singlet superconducting correlations dominate, while for U>Us, the system is in the spin-density wave phase as in the usual Hubbard model. For intermediate values Uc<U<Us, the system is in a spontaneously dimerized bond-ordered wave phase, which is absent in the ordinary Hubbard model with X=0. We obtain that the charge transition remains at Uc=0 for X≠0. Solving the RG equations for the spin sector, we provide an analytical expression for Us(X). The results, with only one adjustable parameter, are in excellent agreement with numerical ones for X<t/2 where t is the hopping.