We report the existence of a regime for domain-wall motion in uniaxial and near-uniaxial ferromagnetic nanowires, characterized by applied magnetic fields sufficiently strong that one of the domains becomes unstable. There appears a stable solution of the Landau-Lifshitz-Gilbert equation, describing a nonplanar domain wall moving with constant velocity and precessing with constant frequency. Even in the presence of thermal noise, the solution can propagate for distances on the order of 500 times the field-free domain-wall width before fluctuations in the unstable domain become appreciable.