It is now well established that a Josephson junction made from conventional spin-singlet superconductors containing ferromagnetic layers can carry spin-triplet supercurrent under certain conditions. The first experimental signature of that fact is the propagation of such supercurrent over long distances through strong ferromagnetic materials. Surprisingly, one of the most salient predictions of the theory has yet to be verified experimentally-namely, that a Josephson junction containing three magnetic layers with coplanar magnetizations should exhibit a ground-state phase shift of either zero or π depending on the relative orientations of those magnetizations. We demonstrate this property using Josephson junctions containing three different types of magnetic layers, chosen so that the magnetization of one layer can be switched by 180° without disturbing the other two. Phase-sensitive detection is accomplished using a superconducting quantum interference device, or SQUID. Such a phase-controllable junction could be used as the memory element in a fully superconducting computer.