Atomic matter waves, like electromagnetic waves, can be focused, reflected, guided and split by currently available passive atom-optical elements. However, the key for many applications of electromagnetic waves lies in the availability of amplifiers. These active devices allow small signals to be detected, and led to the development of masers and lasers. Although coherent atomic beams have been produced1,2,3,4, matter wave amplification has not been directly observed. Here we report the observation of phase-coherent amplification of atomic matter waves. The active medium is a Bose–Einstein condensate, pumped by light that is far off resonance. An atomic wave packet is split off the condensate by diffraction from an optical standing wave, and then amplified. We verified the phase coherence of the amplifier by observing interference of the output wave with a reference wave packet. This development provides a new tool for atom optics and atom interferometry, and opens the way to the construction of active matter-wave devices.