The Super Cryogenic Dark Matter Search (SuperCDMS) experiment is a direct detection dark matter experiment intended for deployment to the SNOLAB underground facility in Ontario, Canada. With a payload of up to 186 germanium and silicon crystal detectors operating below 15 mK, the cryogenic architecture of the experiment is complex. Further, the requirement that the cryostat presents a low radioactive background to the detectors limits the materials and techniques available for construction, and heavily influences the design of the cryogenics system. The resulting thermal architecture is a closed cycle (no liquid cryogen) system, with stages at 50 and 4 K cooled with gas and fluid circulation systems and stages at 1 K, 250 mK and 15 mK cooled by the lower temperature stages of a large, cryogen-free dilution refrigerator. This paper describes the thermal design of the experiment, including details of the cooling systems, mechanical designs and expected performance of the system under operational conditions.