Conditional frequentist tests of a precise hypothesis versus a composite alternative have recently been developed, and have been shown to be equivalent to conventional Bayes tests in the very strong sense that the reported frequentist error probabilities equal the posterior probabilities of the hypotheses. These results are herein extended to sequential testing, and yield fully frequentist sequential tests that are considerably easier to use than are conventional sequential tests. Among the interesting properties of these new tests is the lack of dependence of the reported error probabilities on the stopping rule, seeming to lend frequentist support to the Stopping Rule Principle. Postscript File (440kB)