This paper presents an implemented algorithm that automatically designs fixtures and assembly pallets to hold three-dimensional parts. The designed fixtures rigidly constrain and locate the part, obey task constraints, are robust to part shape variations, are easy to load, and are economical to produce. The algorithm is guaranteed to find the global optimum solution that satisfies these and other pragmatic conditions. We present the results of the algorithm applied to several practical manufacturing problems. For these complex problems the algorithm typically returns initial high-quality fixture designs in less than two minutes, and identifies the global optimum design in just over an hour.