AbstractThis paper treats of a model of a population whose individuals diffuse on a line according to the usual Brownian motion scheme (with constant diffusion coefficient σ22) and at the same time undergo births and deaths with constant rates λ and μ respectively. The “backward” equation governing the transition probability of such a process and an iterative expression for its solution are given. The main results concern the asymptotic spatial distribution of the population conditional on its size as time increases: if the process starts with a single individual at the origin, and if at time t the population has exactly n members in positions y1, …, yn, then the mean position ȳ = ∑1nyin has asymptotically a Gaussian distribution with zero mean and standard deviation σ √t, while the spatial dispersion s2 = ∑1n (yi − ȳ)2 about ȳ has a limiting non-degenerate distribution. Thus for fixed n the mean position gets more diffuse but the dispersion does not grow without limit.