Rectangular current pulses of duration 0.14 mus, flowing across Bloch domain walls in Ni81Fe19 films, cause displacements DELTAx of these walls, observable by Kerr-contrast microscopy. In zero magnetic field, DELTAx reaches congruent-to 14 mum/pulse at current densities congruent-to 30% above the value j(c) where wall motion starts. This critical current density is j(c) congruent-to 1.2 x 10(10) A/m2 for a film thickness w = 263 nm. We have measured j(c) versus film thickness for w = 120-740 nm, and find j(c) is-proportional-to w-2.1. This suggests strongly that the observed wall motion is associated with an S-shaped distortion of the wall by the circumferential magnetic field of the current. This wall distortion is limited by the wall surface tension. The wall structure becomes that of the so-called asymmetric Neel wall. Through wall distortion, the current pulse pumps kinetic energy and momentum into the wall. This kinetic energy is then dissipated during ballistic wall motion happening largely after the end of the pulse. We also find j(c) to be independent of pulse duration.