Electron spectra of photoexcited Ne clusters are shown to display a signal at low kinetic energies that is neither present in the Ne monomer nor at photon energies below the inner-valence 2s threshold. These findings are strong evidence for the existence of interatomic Coulombic decays (ICD), a mechanism that was recently predicted theoretically [Phys. Rev. Lett. 79, 4778 (1997)]]. In ICD, an inner-valence hole state in a weakly bonded system can undergo ultrafast relaxation due to energy transfer to a neighboring atom, followed by electron emission from this neighboring site.