The cryogenic design and performance of an apparatus used to magnetically confine ultracold neutrons (UCN) is presented. The apparatus is part of an effort to measure the beta-decay lifetime of the free neutron and is comprised of a high-current superconducting magnetic trap that surrounds 21 l of isotopically pure 4He cooled to approximately 250 mK. A 0.89 nm neutron beam can enter the apparatus from one end of the magnetic trap and a light collection system allows visible light generated within the helium by decays to be transported to detectors at room temperature. Two cryocoolers are incorporated to reduce liquid helium consumption