We study the properties of the diffuse light in galaxy clusters forming in a large hydrodynamical cosmological simulation of the Λ cold dark matter cosmology. The simulation includes a model for radiative cooling, star formation in dense cold gas, and feedback by Type II supernova explosions. We select clusters having mass M>1014 h-1 Msolar and study the spatial distribution of their star particles. While most stellar light is concentrated in gravitationally bound galaxies orbiting in the cluster potential, we find evidence for a substantial diffuse component, which may account for the extended halos of light observed around central cD galaxies. We find that more massive simulated clusters have a larger fraction of stars in the diffuse light than the less massive ones. The intracluster light is more centrally concentrated than the galaxy light, and the stars in the diffuse component are on average older than the stars in cluster galaxies, supporting the view that the diffuse light is not a random sampling of the stellar population in the cluster galaxies. We thus expect that at least ~10% of the stars in a cluster may be distributed as intracluster light, largely hidden thus far because of its very low surface brightness.