Abstract The main problem to be solved here may be described as follows: let X 1, X2, …, be independent random variables, each with density f 8(x) =1/θ over (0, θ) and zero elsewhere. It is desired to use a two-stage sequential procedure to estimate the unknown parameter θ by an interval of length at most d units and with confidence at least I—α, for some specified d>0 and α in (0, 1). The procedure proposed in this article satisfies an admissibility criterion which may be stated in terms of the maximum possible number of observations or, alternatively, the expected number of observations.