Synchrotrons have for decades provided invaluable sources of soft X-rays, the application of which has led to significant progress in many areas of science and technology. But future applications of soft X-rays--in structural biology, for example--anticipate the need for pulses with much shorter duration (femtoseconds) and much higher energy (millijoules) than those delivered by synchrotrons. Soft X-ray free-electron lasers should fulfil these requirements but will be limited in number; the pressure on beamtime is therefore likely to be considerable. Laser-driven soft X-ray sources offer a comparatively inexpensive and widely available alternative, but have encountered practical bottlenecks in the quest for high intensities. Here we establish and characterize a soft X-ray laser chain that shows how these bottlenecks can in principle be overcome. By combining the high optical quality available from high-harmonic laser sources (as a seed beam) with a highly energetic soft X-ray laser plasma amplifier, we produce a tabletop soft X-ray femtosecond laser operating at 10 Hz and exhibiting full saturation, high energy, high coherence and full polarization. This technique should be readily applicable on all existing laser-driven soft X-ray facilities.