We demonstrate all-optical quantum interference injection and control of a ballistic pure spin current (without an accompanying charge current) in GaAs/AlGaAs quantum wells, consisting of spin-up electrons traveling in one direction and spin-down electrons traveling in the opposite direction. This current is generated through quantum interference of one- and two-photon absorption of approximately 100 fs phase-locked pulses that have orthogonal linear polarizations. We use a spatially resolved pump-probe technique to measure carrier movement of approximately 10 nm. Results agree with recent theoretical predictions.