Comets are icy bodies that sublimate and become active when close to the Sun. They are believed to originate in two cold reservoirs beyond the orbit of Neptune: the Kuiper Belt (equilibrium temperatures of ~40 kelvin) and the Oort Cloud (~10 kelvin). We present optical data showing the existence of a population of comets originating in a third reservoir: the main asteroid belt. The main-belt comets are unlike the Kuiper Belt and Oort Cloud comets in that they likely formed where they currently reside and may be collisionally activated. The existence of the main-belt comets lends new support to the idea that main-belt objects could be a major source of terrestrial water.