Heralded noiseless amplification based on single-photon sources and linear optics is ideally suited for long-distance quantum communication tasks based on discrete variables. We experimentally demonstrate such an amplifier, operating at telecommunication wavelengths. Coherent amplification is performed with a gain of for a state with a maximum expected gain . We also demonstrate that there is no need for a stable phase reference between the initial signal state and the local auxiliary photons used by the amplifier. We discuss these results in the context of experimental device-independent quantum key distribution based on heralded qubit amplification, and we highlight several key challenges for its realization.