Dynamical behavior of solitary waves in the form of kinks in a one dimensional driven damped ferromagnetic spin system is analysed. The Landau-Lifshitz-Gilbert equation for this system is solved exactly. A balance between the effect of the drive field and that of the damping is found to be possible: stationary motion of the kink results with a constant velocity which is proportional to the product of the external magnetic field and of the Gilbert damping constant. The motion of the kink is accompanied by a constant and uniform precession of the spins.