The AEgIS experiment is an international collaboration with the main goal of performing the first direct measurement of the Earth's gravitational acceleration on antimatter. Critical to the success of AEgIS is the production of cold antihydrogen (View the MathML source) atoms. The FACT detector is used to measure the production and temperature of the View the MathML source atoms and for establishing the formation of a View the MathML source beam. The operating requirements for this detector are very challenging: it must be able to identify each of the thousand or so annihilations in the 1 ms period of pulsed View the MathML source production, operate at 4 K inside a 1 T solenoidal field and not produce more than 10 W of heat. The FACT detector consists of two concentric cylindrical layers of 400 scintillator fibres with a 1 mm diameter and a 0.6 mm pitch. The scintillating fibres are coupled to clear fibres which transport the scintillation light to 800 silicon photomultipliers. Each silicon photomultiplier signal is connected to a linear amplifier and a fast discriminator, the outputs of which are sampled continuously by Field Programmable Gate Arrays (FPGAs). In the course of the developments for the FACT detector we have established the performance of scintillating fibres at 4 K by means of a cosmic-ray tracker operating in a liquid helium cryostat. The FACT detector was installed in the AEgIS apparatus in December 2012 and will be used to study the View the MathML source formation when the low energy antiproton physics programs resume at CERN in the Summer of 2014. This paper presents the design requirements and construction methods of the FACT detector and provides the first results of the detector commissioning.