We report a novel depth dependence for the penetration of spin current into ultrathin ferromagnets. Ferromagnetic resonance measurements show that transverse spin current pumped into three structurally distinct ferromagnets is attenuated, on reflection, by an amount proportional to the ferromagnetic layer thickness, saturating abruptly at 1.2 ± 0.1 nm. The observed power-law decay, differing significantly from the (exponential) characteristic-length dependence for longitudinal spin current, confirms models of spin momentum transfer which have been inaccessible to experiment.