A high-gain harmonic-generation free-electron laser is demonstrated. Our approach uses a laser-seeded free-electron laser to produce amplified, longitudinally coherent, Fourier transform-limited output at a harmonic of the seed laser. A seed carbon dioxide laser at a wavelength of 10.6 micrometers produced saturated, amplified free-electron laser output at the second-harmonic wavelength, 5.3 micrometers. The experiment verifies the theoretical foundation for the technique and prepares the way for the application of this technique in the vacuum ultraviolet region of the spectrum, with the ultimate goal of extending the approach to provide an intense, highly coherent source of hard x-rays.