Distortions of the observed cosmic microwave background provide a direct measurement of the microwave background temperature at redshifts from 0 to 1 (refs. 1,2). Some additional background temperature estimates exist at redshifts from 1.8 to 3.3 based on molecular and atomic line-excitation temperatures in quasar absorption-line systems, but are model dependent3. No deviations from the expected (1 + z) scaling behaviour of the microwave background temperature have been seen4, but the measurements have not extended deeply into the matter-dominated era of the Universe at redshifts z > 3.3. Here we report observations of submillimetre line absorption from the water molecule against the cosmic microwave background at z = 6.34 in a massive starburst galaxy, corresponding to a lookback time of 12.8 billion years (ref. 5). Radiative pumping of the upper level of the ground-state ortho-H2O(110-101) line due to starburst activity in the dusty galaxy HFLS3 results in a cooling to below the redshifted microwave background temperature, after the transition is initially excited by the microwave background. This implies a microwave background temperature of 16.4-30.2 K (1σ range) at z = 6.34, which is consistent with a background temperature increase with redshift as expected from the standard ΛCDM cosmology4.