We study the anisotropic Heisenberg (XYZ) spin-1/2 chain placed in a magnetic field pointing along the x axis. We use bosonization and a renormalization group analysis to show that the model has a nontrivial fixed point at a certain value of the XY anisotropy a and the magnetic field h. Hence there is a line of critical points in the (a,h) plane on which the system is gapless, even though the Hamiltonian has no continuous symmetry. The quantum critical line corresponds to a spin-flip transition; it separates two gapped phases in one of which the Z(2) symmetry of the Hamiltonian is broken. Our study has a bearing on one of the transitions of the axial next-nearest neighbor Ising chain in a transverse magnetic field. We also discuss the properties of the model when the magnetic field is increased further, in particular, the disorder line on which the ground state is a direct product of single spin states.