Let $M$ be an asymptotically flat 3-manifold of nonnegative scalar curvature. The Riemannian Penrose Inequality states that the area of an outermost minimal surface $N$ in $M$ is bounded by the ADM mass $m$ according to the formula $|N|\le 16\pi m^2$. We develop a theory of weak solutions of the inverse mean curvature flow, and employ it to prove this inequality for each connected component of $N$ using Geroch's monotonicity formula for the ADM mass. Our method also proves positivity of Bartnik's gravitational capacity by computing a positive lower bound for the mass purely in terms of local geometry.