Domain wall motion induced by nanosecond current pulses in nanostripes with perpendicular magnetic anisotropy (Pt/Co/AlO$_x$) is shown to exhibit negligible inertia. Time-resolved magnetic microscopy during current pulses reveals that the domain walls start moving, with a constant speed, as soon as the current reaches a constant amplitude, and no or little motion takes place after the end of the pulse. The very low mass of these domain walls is attributed to the combination of their narrow width and high damping parameter $alpha$. Such a small inertia should allow accurate control of domain wall motion, by tuning the duration and amplitude of the current pulses.
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