In magnetically anisotropic materials a driving force for grain boundary migration can be induced by an external magnetic field. It is experimentally shown that annealing of locally deformed Zn single crystals in a suitably directed high magnetic field results in a growth of new individual grains. Velocities of some solitary moving grain boundaries were measured and their absolute mobilities were estimated at a single temperature. Results are discussed in terms of preferential grain orientation and boundary character.