Summary We consider sequential tests to decide whether the difference between the means of two normal populations is negative or positive. A Bayesian formulation is adopted, with costs of sampling which differ for the two populations and depend on the difference between the means. The asymptotic “shape”, in the sense of Schwarz (1962), of the Bayes continuation region is found, and this is used to derive an asymptotically Bayes test. A number of asymptotic optimality properties of this test are proved, and it is compared by computer simulation with a two-population SPRT and a truncated generalized two-population SPRT.