The terahertz and far-infrared band, ranging from approximately 0.3 THz to 15 THz (1 mm to 20 mu m), is important for astrophysics as it hosts the peak of the thermal radiation of the cold component of the Universe as well as many spectral lines that trace the cycle of interstellar matter1-8. However, water vapour makes the terrestrial atmosphere opaque to this frequency band over nearly all of the Earth's surface9. Early radiometric measurements10 below 1 THz at Dome A (80 degrees 22'S, 77 degrees 21' E), the highest point of the cold and dry Antarctic ice sheet, suggest that this site may offer the best possible access for ground-based astronomical observations in the terahertz and far-infrared band. To fully assess the site conditions and to address the uncertainties in radiative transfer modelling of the atmosphere, we carried out measurements of atmospheric radiation from Dome A with a Fourier transform spectrometer, spanning the entire water vapour pure rotation band from 20 mu m to 350 mu m. Our measurements reveal substantial transmission in atmospheric windows throughout the whole band. By combining our broad-band spectra with data on the atmospheric state over Dome A, we set new constraints on the spectral absorption of water vapour at upper tropospheric temperatures, which is important for accurate modelling of the terrestrial climate. We find that current spectral models significantly underestimate the H2O continuum absorption.