Summary: Let \(F\) be the gamma distribution function with parameters \(a>0\) and \(\alpha>0\) and let \(G_ s\) be the negative binomial distribution function with parameters \(\alpha\) and \(a/s\), \(s>0\). By combining both probabilistic and approximation-theoretic methods, we obtain sharp upper and lower bounds for \(\sup_{\theta\geq 0} | G_ s(s\theta)- F(\theta)|\), as \(s\to\infty\). In particular, we show that the exact order of uniform convergence is \(s^{-p}\), where \(p=\min(1,\alpha)\). Various kinds of applications concerning charged multiplicity distributions, the Yule birth process and Bernstein-type operators are also given.