The current-induced motion of magnetic domain walls confined to nanostructures is of interest for applications in magnetoelectronic devices in which the domain wall serves as the logic gate or memory element. The injection of spin-polarized current below a threshold value through a domain wall confined to a pinning potential results in its precessional motion within the potential well. We show that by using a short train of current pulses, whose length and spacing are tuned to this precession frequency, the domain wall's oscillations can be resonantly amplified. This makes possible the motion of domain walls with much reduced currents, more than five times smaller than in the absence of resonant amplification.