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 380g. 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 250kgd of data have been collected. This paper describes the technical details of the experimental setup and the hardware components.