We develop a first-principles theory of phonon-assisted optical absorption in semiconductors and insulators which incorporates the temperature dependence of the electronic structure. We show that the Hall-Bardeen-Blatt theory of indirect optical absorption and the Allen-Heine theory of temperature-dependent band structures can be derived from the present formalism by retaining only one-phonon processes. We demonstrate this method by calculating the optical absorption coefficient of silicon using an importance sampling Monte Carlo scheme, and we obtain temperature-dependent lineshapes and band gaps in good agreement with experiment. The present approach opens the way to predictive calculations of the optical properties of solids at finite temperature. ; Comment: 11 pages, 9 figures