Submonolayer quantities of CH3I have been adsorbed onto an atomically-clean Cu(111) single-crystal surface under ultra-high-vacuum conditions. At 110 K, the molecules remain intact. Upon heating the surface, the carbon-iodine bond dissociates at 140 K to produce adsorbed methyl groups (which have been identified by surface vibrational spectroscopy) and gas-phase methyl radicals (which are detected by mass spectrometry). The activation energy for these processes is approximately 8.5 kcal/mol. The observation of methyl radicals suggests that the surface reaction is analogous to the oxidative addition of alkyl iodides to metal compounds by atom-transfer or electron-transfer mechanisms. The thermodynamics and kinetics of both mechanisms are discussed in the context of the Cu(111) results.