We may take observations sequentially from a population with unknown mean \theta. After this sampling stage, we are to decide whether \theta is greater or less than a known constant \nu. The net worth upon stopping is either \theta or \nu, respectively, minus sampling costs. The objective is to maximize the expected net worth when the probability measure of the observations is a Dirichlet process with parameter \alpha. The stopping problem is shown to be truncated when \alpha has bounded support. The main theorem of the paper leads to bounds on the exact stage of truncation and shows that sampling continues longest on a generalized form of neutral boundary.