The First G-APD Cherenkov Telescope (FACT) is designedto detect cosmic gamma-rays with energies from several hundred GeV upto about 10 TeV using the Imaging Atmospheric Cherenkov Technique. Incontrast to former or existing telescopes, the camera of the FACTtelescope is comprised of solid-state Geiger-mode AvalanchePhotodiodes (G-APD) instead of photomultiplier tubes for photodetection. It is the first full-scale device of its kind employingthis new technology. The telescope is operated at the Observatorio delRoque de los Muchachos (La Palma, Canary Islands, Spain) since fall2011. This paper describes in detail the design, construction andoperation of the system, including hardware and softwareaspects. Technical experiences gained after one year of operation arediscussed and conclusions with regard to future projects are drawn.