Proteins play important roles in biology, biotechnology and pharmacology, and missense variants are a common cause of disease. Discovering functionally important sites in proteins is a central but difficult problem because of the lack of large, systematic data sets. Sequence conservation can highlight residues that are functionally important but is often convoluted with a signal for preserving structural stability. We here present a machine learning method to predict functional sites by combining statistical models for protein sequences with biophysical models of stability. We train the model using multiplexed experimental data on variant effects and validate it broadly. We show how the model can be used to discover active sites, as well as regulatory and binding sites. We illustrate the utility of the model by prospective prediction and subsequent experimental validation on the functional consequences of missense variants in HPRT1 which may cause Lesch-Nyhan syndrome, and pinpoint the molecular mechanisms by which they cause disease.