The chemical dynamics of highly vibrationally excited molecules have been studied by measuring the quantum state-resolved scattering probabilities of nitric oxide (NO) molecules on clean and oxygen-covered copper (111) surfaces, where the incident NO was prepared in single quantum states with vibrational energies of as much as 300 kilojoules per mole. The dependence of vibrationally elastic and inelastic scattering on oxygen coverage strongly suggests that highly excited NO (v = 13 and 15) reacts on clean copper (111) with a probability of 0.87 +/- 0.05, more than three orders of magnitude greater than the reaction probability of ground-state NO. Vibrational promotion of surface chemistry on metals (up to near-unit reaction probability) is possible despite the expected efficient relaxation of vibrational energy at metal surfaces.