A population of particles is considered whose size X_N(t) changes according to a branching stochastic process. The purpose of this paper is to find an approximate distribution of X_N(t) when t is fixed (not necessarily large) but the initial size of the population, N, is large. If N is allowed to tend to infinity, and if the parameters of the process are made to change in a way analogous to the Poisson approximation of a binomial distribution, then it is shown that a limiting distribution of the process X_N(t) - N exists as N \rightarrow \infty, and this limiting distribution is the distribution of a continuous process with independent increments. The relation between the parameters of the infinitely divisible distribution of the limiting process and the sequence of branching processes is exhibited.