Let A be a partial positive definite Hermitian matrix, and let G(A) be the undirected graph of the specified off-diagonal entries of A. When G(A) is a chordal graph, we present an explicit determinantal formula for the (unique) determinant-maximizing positive definite completion (maximum-entropy completion) of A. In order to do this, we first give a characterization of a chordal graph in terms of the existence of a proper spanning tree T of the intersection graph of its set of maximal cliques. The formula for the maximum determinant is then a quotient of products of specified principal minors of A, in which each minor in the numerator corresponds to a maximal clique of G(A) while each minor in the denominator corresponds to the intersection of two maximal cliques associated with an edge of T. We next show that the denominator terms can be identified graph-theoretically with the minimal vertex separators of G(A), and that these terms are independent of the particular proper spanning tree T. Then an alternative formula is given for the maximum determinant as an “inclusion-exclusion”-like ratio of principal minors associated with the maximal cliques of G(A). Finally, we show that the entries of the determinant-maximizing completion may be calculated by solving a sequence of simple one-variable maximization problems.