The solution of the problem of the relative stability of all possible equilibrium bubble states that can form from a closed, finite, supersaturated gas-liquid solution, maintained at a fixed temperature and a fixed external pressure is given. The supersaturated solution may contain any number of dissolved volatile solutes. The full solution to this problem has remained elusive for decades, because of the complication of pressure inequalities between the bubbles and the constant external (or reservoir) pressure. The method of solution is one that had been used previously to solve the related problem of the stability of a liquid droplet in a supersaturated vapor, where the same complication occurred. The derived equations were found to reduce correctly when simplified; they were consistent with experiment, and the system Gibbs free energy appropriately obeyed the Law of Corresponding States.. The expressions were used in the context of transition state theory to provide semiempirical predictions of the rate of homogeneous bubble formation from a supersaturated solution, and the "critical pressure for homogeneous nucleation (P-crit)". The nucleation Gibbs free energy expression derived here had a lower barrier height and resulted in a reduction of P-crit values, relative to what was obtained from the basis of a pre-existing approximate expression taken from the literature. Applications to chemical engineering and human decompression modeling are briefly described.