The COBRA demonstrator, a prototype for a large-scale experiment searching for neutrinoless double beta-decay, was built at the underground laboratory Laboratori Nazionali del Gran Sasso (LNGS) in Italy. It consists of an array of 64 monolithic, calorimetric CdZnTe semiconductor detectors with a coplanar-grid design and a total mass of 380 g. It is used to investigate the experimental challenges faced when operating CdZnTe detectors in low-background mode, to identify potential background sources and to show the long-term stability of the detectors. The first data-taking period started in 2011 with a subset of the detectors, while the demonstrator was completed in November 2013. To date, more than 250 kg d of data have been collected. This paper describes the technical details of the experimental setup and the hardware components.