Geiger-mode Avalanche Photodiodes (G-APD) bear the potential to significantly improve the sensitivity of Imaging Air Cherenkov Telescopes (IACT). We are currently building the First G-APD Cherenkov Telescope (FACT) by refurbishing an old IACT with a mirror area of 9.5 square meters and are constructing a new, fine-pixelized camera using novel G-APDs. The main goal is to evaluate the performance of a complete system by observing very high energy gamma-rays from the Crab Nebula. This is an important field test to check the feasibility of G-APD-based cameras to replace at some time the PMT-based cameras of planned future IACTs like AGE and CTA. In this article, we present the basic design of such a camera as well as some important details. (C) 2010 Elsevier B.V. All rights reserved.