Within the FACT project, we construct a new type of camera based on Geiger-mode avalanche photodiodes (G-APDs). Compared to photomultipliers, G-APDs are more robust, need a lower operation voltage and have the potential of higher photon-detection efficiency and lower cost, but were never fully tested in the harsh environments of Cherenkov telescopes. The FACT camera consists of 1440 G-APD pixels and readout channels, based on the DRS4 (Domino Ring Sampler) analog pipeline chip and commercial Ethernet components. Preamplifiers, trigger system, digitization, slow control and power converters are integrated into the camera.