Since the invention of the laser more than 50 years ago, scientists have striven to achieve amplification on atomic transitions of increasingly shorter wavelength. The introduction of X-ray free-electron lasers makes it possible to pump new atomic X-ray lasers with ultrashort pulse duration, extreme spectral brightness and full temporal coherence. Here we describe the implementation of an X-ray laser in the kiloelectronvolt energy regime, based on atomic population inversion and driven by rapid K-shell photo-ionization using pulses from an X-ray free-electron laser. We established a population inversion of the Kα transition in singly ionized neon at 1.46 nanometres (corresponding to a photon energy of 849 electronvolts) in an elongated plasma column created by irradiation of a gas medium. We observed strong amplified spontaneous emission from the end of the excited plasma. This resulted in femtosecond-duration, high-intensity X-ray pulses of much shorter wavelength and greater brilliance than achieved with previous atomic X-ray lasers. Moreover, this scheme provides greatly increased wavelength stability, monochromaticity and improved temporal coherence by comparison with present-day X-ray free-electron lasers. The atomic X-ray lasers realized here may be useful for high-resolution spectroscopy and nonlinear X-ray studies.