The classic sequential search problem rewards the decision maker with the highest sampled value, minus a cost per sample. If the sampling distribution is unknown, then a Bayesian decision maker faces a complex balance between exploration and exploitation. We solve the stopping problem of sampling from a Normal distribution with unknown mean and unknown variance and a conjugate prior, a longstanding open problem. The optimal stopping region may be empty (it may be optimal to continue the search regardless of the offer one receives, especially at the early stages), or it may consist of one or two bounded intervals. While a single reservation price cannot describe the optimal rule, we do find a standardized reservation rule: stop if and only if the standardized value of the current offer is sufficiently high relative to the standardized search cost. We also introduce the index function, which provides a computationally practical way to implement the standardized stopping rule for any given prior, sampling history, and sampling horizon.