Maximum likelihood estimation of the parameter λ of a pure birth process is studied on the assumptions that the process is observed either continuously in a time interval [0, t] or at equidistant time points O, T, ..., KT. The exact distribution of the minimal sufficient statistic is given in the first case and for both cases the asymptotic theory as t→ ∞, or as, k →, ∞, is studied. The related conditional Poisson process discussed recently by D. G. Kendall and W. A. O'N. Waugh is also studied, and the results are shown to illustrate the modern theory of exponential families and conditional inference.