It is well established that colloidal particles adsorbed on bubble surfaces can increase bubble and foam lifetimes by several orders of magnitude in gas-saturated solutions. Nevertheless, in spite of the many reports of long-lived foams and bubbles covered with particles (armored bubbles), the mechanism for the stabilization of armored bubbles remains an open question. Here we address the issue of stabilization using both experimental and numerical approaches. In our experiments, we find that initially spherical armored bubbles deform into nonspherical shapes as they stabilize. In gas-saturated solutions, these shapes are characterized by planar facets or folds for decreasing ratios of the particle to bubble radii. We perform Surface Evolver simulations that mimic armored bubble dissolution, and find that the faceted shape represents a local minimum of energy of the system. Furthermore, we find that the faceted shape also corresponds to the vanishing of the Laplace overpressure. Our simulations reveal that the vanishing of the Laplace overpressure is due to the saddle-shape deformation of most of the interface as the bubble facets. We propose that this deformation-induced elimination of the Laplace overpressure is the mechanism of stabilization of armored bubbles.