The clique-separator graph is a new representation of chordal graphs that provides more information about a graph's structure than the well known clique tree representation. Moreover, when a chordal graph is interval, proper interval, or split, its clique-separator graph has additional structural properties. The clique-separator graph leads to new characterizations of proper interval graphs and split graphs, and it may lead to analogous results for other subclasses of chordal graphs.