We report conductance measurements of ballistic one-dimensional (1D) wires defined in GaAs/AlGaAs heterostructures in an in-plane magnetic field, B. When the Zeeman energy is equal to the 1D subband energy spacing, the spin-split subband N(up-spin) intersects (N+1)(down-spin), where N is the index of the spin-degenerate 1D subband. At the crossing of N=1(up-spin) and N=2(down-spin) subbands, there is a spontaneous splitting giving rise to an additional conductance structure evolving from the 1.5(2e^2/h) plateau. With further increase in B, the structure develops into a plateau and lowers to 2e^2/h. With increasing temperature and magnetic field the structure shows characteristics of the 0.7 structure. Our results suggest that at low densities a spontaneous spin splitting occurs whenever two 1D subbands of opposite spins cross.