Co/Pt multilayer films with perpendicular magnetic anisotropy and large out-of-plane coercivities of 3.9 - 8.5 kOe have been found to undergo a spin reorientation transition from out-of-plane to in-plane upon irradiation with 700 keV nitrogen ions. X-ray reflectivity experiments show that the multilayer structure gets progressively disrupted with increasing ion dose, providing direct evidence for local atomic displacements at the Co/Pt interfaces. This effectively destroys the magnetic interface anisotropy, which was varied by about a factor of two, between KS@ 0.4 erg/cm2 and KS@ 0.85 erg/cm2 for two particular films. The dose required to initiate spin-reorientation, 6x1014 N+/cm2 and 1.5x1015 N+/cm2, respectively, scales with KS. It is roughly equal to the number of Co interface atoms per unit interface area contributing to KS.