Insulating antiferromagnets have recently emerged as efficient and robust conductors of spin current. Element-specific and phase-resolved x-ray ferromagnetic resonance has been used to probe the injection and transmission of ac spin current through thin epitaxial NiO(001) layers. The spin current is found to be mediated by coherent evanescent spin waves of GHz frequency, rather than propagating magnons of THz frequency, paving the way towards coherent control of the phase and amplitude of spin currents within an antiferromagnetic insulator at room temperature.