Recent observations, including the discovery in typical asteroidal orbits of objects with cometary characteristics (main-belt comets, or MBCs), have blurred the line between comets and asteroids, although so far neither ice nor organic material has been detected on the surface of an asteroid or directly proven to be an asteroidal constituent. Here we report the spectroscopic detection of water ice and organic material on the asteroid 24 Themis, a detection that has been independently confirmed. 24 Themis belongs to the same dynamical family as three of the five known MBCs, and the presence of ice on 24 Themis is strong evidence that it also is present in the MBCs. We conclude that water ice is more common on asteroids than was previously thought and may be widespread in asteroidal interiors at much smaller heliocentric distances than was previously expected.