By using an Edgeworth expansion to speed up the asymptotics, we develop one-sided coverage intervals for a proportion based on a stratified simple random sample that work better than well-known competitors. Unlike in the simple random sampling case, it is necessary to estimate the variance of the estimator for the population proportion when the stratum means are allowed to differ. The idealized variance estimator we employ, although not perfect, appears to be useful. Its use with the Edgeworth expansion leads to coverage-interval applications beyond proportions estimated from stratified random samples.