We study the effects of spin-orbit interaction (SOI) on the current-induced motion of a magnetic ( Bloch) domain wall in ultrathin ferromagnetic nanowires. The conspiracy of spin relaxation and SOI is shown to generate a strong nonequilibrium Rashba field, which can dominate even for weak SOI. This field causes intricate spin precession and a transition from translatory to oscillatory wall dynamics with increasing SOI. We show that current pulses of different lengths can be used to efficiently control the domain wall motion.