The assumption of log-concavity is a flexible and appealing non-parametric shape constraint in distribution modelling. In this work, we study the log-concave maximum likelihood estimator of a probability mass function. We show that the maximum likelihood estimator is strongly consistent and we derive its pointwise asymptotic theory under both the well-specified and misspecified settings. Our asymptotic results are used to calculate confidence intervals for the true log-concave probability mass function. Both the maximum likelihood estimator and the associated confidence intervals may be easily computed by using the R package logcondiscr. We illustrate our theoretical results by using recent data from the H1N1 pandemic in Ontario, Canada.