We consider sequences of metrics, $g_j$, on a Riemannian manifold, $M$, which converge smoothly on compact sets away from a singular set $ S \subset M$, to a metric, $g_\infty$, on $M ∖setminus S$. We prove theorems which describe when $M_j=(M,g_j) $converge in the Gromov-Hausdorff sense to the metric completion, $(M_\infty,d_\infty), of $(M∖setminus S,g_\infty)$. To obtain these theorems, we study the intrinsic flat limits of the sequences. A new method, we call hemispherical embedding, is applied to obtain explicit estimates on the Gromov-Hausdorff and Intrinsic Flat distances between Riemannian manifolds with diffeomorphic subdomains.