The longitudinal space-charge amplifier has been recently proposed by Schneidmiller and Yurkov as an alternative to the free-electron laser instability for the generation of intense broadband radiation pulses [Phys. Rev. ST Accel. Beams 13, 110701 (2010)]. In this Letter, we report on the experimental
demonstration of a cascaded longitudinal space-charge amplifier at optical wavelengths. Although seeded by electron beam shot noise, the strong compression of the electron beam along the three amplification stages leads to emission of coherent undulator radiation pulses exhibiting a single spectral spike and a
single transverse mode. The on-axis gain is estimated to exceed 4 orders of magnitude with respect to spontaneous emission.