An experimental search for dark matter particle candidates using cryogenic detectors requires a low radioactive background environment. We discuss the status of a cryogenic dark matter experiment to be performed in the Stanford Underground Facility. The detectors will be cooled in a specially designed cryostat connected to a modified side access Oxford 400 dilution refrigerator. Details of the cryostat design and its operating performance are presented. The effectiveness of the multi-level shield surrounding the cryostat, as well as the background levels we expect to achieve in the pilot experiment are discussed. Finally, we examine the limits which can be set on dark matter candidates with such an experiment.