  The Positive Mass Conjecture states that any complete asymptotically flat
manifold of nonnnegative scalar curvature has nonnegative mass. Moreover, the
equality case of the Positive Mass Conjecture states that in the above
situation, if the mass is zero, then the Riemannian manifold must be Euclidean
space. The Positive Mass Conjecture was proved by R. Schoen and S.-T. Yau for
all manifolds of dimension less than 8, and it was proved by E. Witten for all
spin manifolds. In this paper, we consider complete asymptotically flat
manifolds of nonnegative scalar curvature that are also harmonically flat in an
end. We show that, whenever the Positive Mass Theorem holds, any appropriately
normalized sequence of such manifolds whose masses converge to zero must have
metrics that are uniformly converging to Euclidean metrics outside a compact
region. This result is an ingredient in a forthcoming proof, co-authored with
H. Bray, of the Riemannian Penrose inequality in dimensions less than 8.
