Department of MathematicsWe prove that topological disks with positive curvature and strictly convex boundary of large length are close to round spherical caps of constant boundary curvature in the Gromov-Hausdorff and Sormani-Wenger Intrinsic Flat senses. This proves stability for a theorem of F. Hang and X. Wang. As an intermediate step we obtain a result concerning gauge fixing and compactness for solutions of a Liouville type PDE.