The approach to the equilibrium (Boltzmann) distribution of populations of internal states of a molecule is governed by inelastic collisions in the gas phase and with surfaces. The set of differential equations governing the time evolution of the internal state populations is commonly called the master equation. An analytic solution to the master equation is presented and shows that the equilibrium distribution is the Boltzmann distribution. This solution is applied to the master equation involving collisions of rotational states of a diatomic molecule with a monatomic bath gas. (Contains 1 table and 5 figures.)