Meta-analysis of proportions is conceptually simple: Faced with a binary outcome in multiple studies, we seek inference on some overall proportion of successes/failures. Under common effect models, exact inference has long been available, but is not when we more realistically allow for heterogeneity of the proportions. Instead a wide range of nonexact fixed-effects methods are used, the interpretation of some of which is challenging. In this paper, we present methods for exact statistical tests and confidence intervals for fixed-effects meta-analysis of proportions. These methods retain the interpretability of the underlying parameter of interest, and can be implemented in straightforward software. We also show how our inference on the overall proportion is compatible with exact inference on heterogeneity of proportions. An illustrative example from a recent kidney disease study shows how the method's performance can be assessed in practice.