We study the C*-closure U of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a compact connected manifold X with boundary partial derivative X not equal phi. We find short exact sequences in K-theory 0 -> K-i (C(X)) -> K-i(U/R) ->(P) K1-i(C-0(T*X degrees)) -> 0, i = 0,1, which split, so that K-i(U/R) congruent to K-i(C(X)) circle plus K1-i(Co(T*X degrees)). Using only simple K-theoretic arguments and the Atiyah-Singer index theorem, we show that the Fredhohn index of an elliptic element in A is given by ind A = ind(t)(p[A])), where [A] is the class of A in K-1(U/R) and ind(t) is the topological index, a relation first established by Boutet de Monvel by different methods.