The performance of an indium-gallium-arsenide avalanche photodiode serving as a 1550 nm single-photon detector is investigated. Quantum efficiency is evaluated for laser pulses with an average of < 0.015 photons per pulse, which are important for the demonstration of unconditionally secure quantum key distribution [G. Brassard et al.: Phys. Rev. Lett. 85, 6, p.1330 (2000)]. An operating temperature of 243 K is achieved by thermo-electrical cooling, yielding a quantum efficiency of 18% with a dark-count probability per gate of 2.8 10-5. The results obtained here guarantee unconditionally secure fiber-optic quantum key distribution over 50 km.