  We 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 sense. This proves stability for a theorem of
F. Hang and X. Wang, and can be viewed as an affirmative answer to a convex
stability version of the Min-Oo Conjecture in dimension two. As an intermediate
step, we obtain a compactness result for a Liouville-type PDE problem.
