Lengths and multiplicities of integrally closed modules over a two-dimensional regular local ring

Let $(R,{\bf m})$ be a two-dimensional regular local ring with infinite residue field. We prove an analogue of the Hoskin-Deligne length formula for a finitely generated, torsion-free, integrally closed $R$-module $M$. As a consequence, we get a formula for the Buchsbaum-Rim multiplicity of $F/M$, where $F = M^{**}$.