When a gas is injected into a bubble in an elastomer, the bubble may first expand gradually, and then snap suddenly to a large size. This snap-through instability is analyzed here using a model that accounts for both the surface tension and the limiting stretch of the elastomer. In a state of equilibrium, the pressure in the bubble counteracts three contributions: the ambient pressure due to the environment outside the elastomer, the Laplace pressure due to the surface tension of the elastomer, and the additional pressure due to the elasticity of the elastomer. The Laplace pressure is large for a small bubble, but falls as the bubble expands. The additional pressure due to elasticity increases as the bubble expands, and rises steeply when the surface of the bubble approaches the limiting stretch of the elastomer. We show that the bubble snaps only if a sufficient amount of gas can rush into the bubble at the onset of instability.