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MUSIC (Multicolor Submillimeter kinetic Inductance Camera) is a new facility instrument for the Caltech Submillimeter Observatory (Mauna Kea, Hawaii) developed as a collaborative effect of Caltech, JPL, the University of Colorado at Boulder and UC Santa Barbara, and is due for initial commissioning in early 2011. MUSIC utilizes a new class of superconducting photon detectors known as microwave kinetic inductance detectors (MKIDs), an emergent technology that offers considerable advantages over current types of detectors for submillimeter and millimeter direct detection. MUSIC will operate a focal plane of 576 spatial pixels, where each pixel is a slot line antenna coupled to multiple detectors through on-chip, lumped-element filters, allowing simultaneously imaging in four bands at 0.86, 1.02, 1.33 and 2.00 mm. The MUSIC instrument is designed for closed-cycle operation, combining a pulse tube cooler with a two-stage Helium-3 adsorption refrigerator, providing a focal plane temperature of 0.25 K with intermediate temperature stages at approximately 50, 4 and 0.4 K for buffering heat loads and heat sinking of optical filters. Detector readout is achieved using semi-rigid coaxial cables from room temperature to the focal plane, with cryogenic HEMT amplifiers operating at 4 K. Several hundred detectors may be multiplexed in frequency space through one signal line and amplifier. This paper discusses the design of the instrument cryogenic hardware, including a number of features unique to the implementation of superconducting detectors. Predicted performance data for the instrument system will also be presented and discussed.