<jats:title>Abstract</jats:title><jats:p>Distortions of the observed cosmic microwave background provide a direct measurement of the microwave background temperature at redshifts from 0 to 1 (refs. <jats:sup>1,2</jats:sup>). 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 dependent<jats:sup>3</jats:sup>. No deviations from the expected (1 + <jats:italic>z</jats:italic>) scaling behaviour of the microwave background temperature have been seen<jats:sup>4</jats:sup>, but the measurements have not extended deeply into the matter-dominated era of the Universe at redshifts <jats:italic>z</jats:italic> > 3.3. Here we report observations of submillimetre line absorption from the water molecule against the cosmic microwave background at <jats:italic>z</jats:italic> = 6.34 in a massive starburst galaxy, corresponding to a lookback time of 12.8 billion years (ref. <jats:sup>5</jats:sup>). Radiative pumping of the upper level of the ground-state ortho-H<jats:sub>2</jats:sub>O(1<jats:sub>10</jats:sub>–1<jats:sub>01</jats:sub>) 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 (1<jats:italic>σ</jats:italic> range) at <jats:italic>z</jats:italic> = 6.34, which is consistent with a background temperature increase with redshift as expected from the standard ΛCDM cosmology<jats:sup>4</jats:sup>.</jats:p>