  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 $F = M^{**}$.
