A cryogenic stopping cell for stopping energetic radioactive ions and extracting them as a low energy beam was developed. This first ever cryogenically operated stopping cell serves as prototype device for the Low-Energy Branch of the Super-FRS at FAIR. The cell has a stopping volume that is 1 m long and 25 cm in diameter. Ions are guided by a DC field along the length of the stopping cell and by a combined RF and DC fields provided by an RF carpet at the exit-hole side. The ultra-high purity of the stopping gas required for optimum ion survival is reached by cryogenic operation. The design considerations and construction of the cryogenic stopping cell, as well as some performance characteristics, are described in detail. Special attention is given to the cryogenic aspects in the design and construction of the stopping cell and the cryocooler-based cooling system. The cooling system allows the operation of the stopping cell at any desired temperature between about 70 K and room temperature. The cooling system performance in realistic on-line conditions at the FRS Ion Catcher Facility at GSI is discussed. A temperature of 110 K at which efficient ion survival was observed is obtained after 10 h of cooling. A minimum temperature of the stopping gas of 72 K was reached. The expertise gained from the design, construction and performance of the prototype cryogenic stopping cell has allowed the development of a final version for the Low-Energy Branch of the Super-FRS to proceed.