Let X be a smooth manifold with boundary of dimension n > 1. The operators of order -n and type zero in Boutet de Monvel's calculus form a subset of Dixmier's trace ideal ℒ^1,∞(H) for the Hilbert space H = L²(X, E) ⊕ L²(∂X, F) of L² sections in vector bundles E over X, F over ∂X. We show that, on these operators, Dixmier's trace can be computed in terms of the same expressions that determine the noncommutative residue. In particular it is independent of the averaging procedure. However, the noncommutative residue and Dixmier's trace are not multiples of each other unless the boundary is empty. As a corollary we show how to compute Dixmier's trace for parametrices or inverses of classical elliptic boundary value problems of the form Pu = f; Tu = 0 with an elliptic differential operator P of order n in the interior and a trace operator T. In this particular situation, Dixmier's trace and the noncommutative residue do coincide up to a factor.