This article addresses the problem of removable singularities for a Hermitian-holomorphic vector bundle ℰ, defined on the complement of an analytic set A of complex codimension at least two in a complex n-dimensional manifold X. In particular it is shown here that there exists a unique holomorphic bundle [Formula: see text] on X, such that [Formula: see text], when the curvature of ℰ belongs to Ln (X\A). This result is in fact sharp, as counterexamples exist for the extensibility of ℰ with curvature in Lp, p < n. Extension across general closed subsets of finite (2n - 4)-dimensional Hausdorff measure then follows directly from a slicing theorem of Bando and Siu.