This paper derives and exhibits the optimum Bayes solution to the following problem: Given a continuous-time Poisson process with unknown mean occurrence rate $\lambda$; to decide whether $\lambda > k$ or $\lambda < k$. The prior distribution is taken to be of Gamma type, with positive mean and finite variance. The cost of observation is taken proportional to the length of time the process is observed, and the cost of a wrong decision proportional to $|\lambda - k|$. The decision rule derived is optimum (in the sense of minimum expected cost) among all non-randomized sequential rules. Some of the results hold true, of course, for other cost functions and/or prior distributions. A method for treating the same problem with the inclusion of a constant setup cost is also given.