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The Levitated Dipole Experiment (LDX) is a novel concept that examines plasma compressibility as a method for stable magnetic confinement of fusion grade plasmas. The experiment uses a 0.8 m diameter ring-type dipole coil that is levitated at the center of a 5 m diameter × 3 m tall vacuum chamber to confine the plasma. This persistent mode, floating coil is wound from a prereacted Nb3Sn conductor and encased in a toroidally shaped, constant volume helium cryostat to eliminate external connections to the coil during levitated operation. Although the peak field on the inductively charged floating coil is only 5.3 T, a Nb3Sn conductor was selected because of its higher temperature capability. The cryostat, with on-board helium supply, is designed for 6-8 hours of levitated operation as the heat leak gradually warms the coil from 5 to 10 K. The cryostat consists of three concentric shells: a sealed, high pressure Inconel helium vessel that contains the floating coil and heat exchangers that are used to recool the coil before operation, a high heat capacity fiberglass-lead radiation shield, and an outer vacuum shell. The shells are kept separated by a support system designed to withstand impact forces up to 10 g in the case of a levitation failure. The paper summarizes the manufacture and initial driven-mode test of the floating coil, and describes the design, manufacture and test of the cryostat.