The decision problem discussed concerns the drift of a Wiener process with known variance per unit time. It is required to find the optimal sequential test for the sign of the drift, when sampling costs and the cost of a wrong terminal decision are specified. Initially, the drift is supposed to have a continuous prior distribution. The optimal procedure is determined by a continuation region in which a certain partial differential equation holds. The boundaries of this region can be found explicitly in a few special cases. In general, methods are developed for obtaining both inner and outer approximations to the optimal continuation region and these techniques are illustrated by application to several examples.