The First G-APD Cherenkov Telescope (FACT) is designed to detect cosmic gamma-rays with energies from several hundred GeV up to about 10 TeV using the Imaging Atmospheric Cherenkov Technique. In contrast to former or existing telescopes, the camera of the FACT telescope is comprised of solid-state Geiger-mode Avalanche Photodiodes (G-APD) instead of photomultiplier tubes for photo detection. It is the first full-scale device of its kind employing this new technology. The telescope is operated at the Observatorio del Roque de los Muchachos (La Palma, Canary Islands, Spain) since fall 2011. This paper describes in detail the design, construction and operation of the system, including hardware and software aspects. Technical experiences gained after one year of operation are discussed and conclusions with regard to future projects are drawn.