A fast algorithm for calculating the nonparametric maximum likelihood estimate (MLE) of a mixing distribution,G(·). in a mixture model is discussed. The literature contains several methods for computing the nonparametric MLE of G, but these methods are slow. In this article we develop an algorithm for maximizing the log-likelihoodl(G) over the family ℊ, of all distribution functions, that yields the nonparametric MLE ofG. In some semiparametric problems, a structural or fixed parameterβis of interest, and we are interested in computing profile likelihood, supGl(G, β), for a grid of values ofβ. The algorithm that we propose is fast enough for this purpose. Examples illustrate and compare the algorithms; one taken from an article by Laird, the common mean problem, and one taken from the literature on optimal experimental design. It is also noted that routines from the literature on semi-infinite programming may be used to compute the profile log-likelihood.