In 1980, the US Department of Energy gave the Lawrence Livermore National Laboratory approval to design and build a tandem Mirror Fusion Test Facility (MFTF-B) to support the goals of the National Mirror Program. We designed the MFTF-B vacuum vessel both to maintain the required ultrahigh vacuum environment and to structurally support the 42 superconducting magnets plus auxiliary internal and external equipment. During our design work, we made extensive use of both simple and complex computer models to arrive at a cost-effective final configuration. As part of this work, we conducted a unique dynamic analysis to study the interaction of the 32,000-tonne concrete-shielding vault with the 2850-tonne vacuum vessel system. To maintain a vacuum of 2 x 10/sup -8/ torr during the physics experiments inside the vessel, we designed a vacuum pumping system of enormous capacity. The vacuum vessel (4200-m/sup 3/ internal volume) has been fabricated and erected, and acceptance tests have been completed at the Livermore site. The rest of the machine has been assembled, and individual systems have been successfully checked. On October 1, 1985, we began a series of integrated engineering tests to verify the operation of all components as a complete system.