A model for gas transport from an encapsulated microbubble into the surrounding medium is developed and investigated incorporating the effects of encapsulation elasticity. Encapsulation elasticity stabilizes microbubbles against dissolution and explains the long shelf life of microbubble contrast agent. We consider air bubbles as well as bubbles containing perfluorocarbon gas. Analytical conditions between saturation level, surface tension and interfacial dilatational elasticity are determined for attaining non-zero equilibrium radius for these microbubbles. Numerical solution of the equation verifies the stability of the equilibrium radii. In an undersaturated medium all encapsulated bubbles dissolve. In a saturated medium, an encapsulated bubble is found to achieve a long-time stable radius when interfacial dilatational elasticity is larger than equilibrium surface tension. For bubbles with interfacial dilatational elasticity smaller than the equilibrium surface tension, stable bubble of non-zero radius can be achieved only when the saturation level is greater than a critical value. Even if they initially contain a gas other than air, bubbles that reach a stable radius finally become air bubbles. The model is applied to an octafluoropropane filled lipid-coated 2.5 microm bubble, which displayed a transient swelling due to air intake before reaching an equilibrium size. Effects of elasticity, shell permeability, initial mole fraction, initial radius and saturation level are investigated and discussed. Shell permeability and mole fraction do not affect the final equilibrium radius of the microbubble but affect the time scale and the transient dynamics. Similarly, the ratio of equilibrium radius to initial radius remains unaffected by the variation in initial radius.