If the Riemann curvature tensor associated with a smooth field C of positive-definite symmetric matrices of order n vanishes in a simply-connected open subset Ω ⊂ R^n, then C is the metric tensor field of a manifold isometrically immersed in R^n. In this Note, we first show how, under a mild smoothness assumption on the boundary of Ω, this classical result can be extended “up to the boundary”. When Ω is bounded, we also establish the continuity of the manifold with boundary obtained in this fashion as a function of its metric tensor field, the topologies being those of the Banach spaces C^k(Ω).