The Leidenfrost effect, where an evaporating liquid or sublimating solid levitates on its own vapor, has been appreciated by scientists for many decades. Here, we demonstrate that an ice disk placed on a hot surface exhibits a composite Leidenfrost effect, where both liquid and vapor films emanate from under the levitating ice. By varying the surface temperature and ice mass, we construct a phase map that shows when the composite Leidenfrost effect is possible. We also estimate the flow rate and height of both the liquid and vapor films, by relating the conductive heat transfer across the films to the latent heat of fusion and vaporization, respectively.