The earliest galaxies are thought to have emerged during the first billion years of cosmic history, initiating the ionization of the neutral hydrogen that pervaded the Universe at this time. Studying this 'epoch of reionization' involves looking for the spectral signatures of ancient galaxies that are, owing to the expansion of the Universe, now very distant from Earth and therefore exhibit large redshifts. However, finding these spectral fingerprints is challenging. One spectral characteristic of ancient and distant galaxies is strong hydrogen-emission lines (known as Lyman-alpha lines), but the neutral intergalactic medium that was present early in the epoch of reionization scatters such Lyman-alpha photons. Another potential spectral identifier is the line at wavelength 157.4 micrometres of the singly ionized state of carbon (the [C II] lambda = 157.74 mu m line), which signifies cooling gas and is expected to have been bright in the early Universe. However, so far Lyman-alpha-emitting galaxies from the epoch of reionization have demonstrated much fainter [C II] luminosities than would be expected from local scaling relations1-5, and searches for the [C II] line in sources without Lyman-a emission but with photometric redshifts greater than 6 (corresponding to the first billion years of the Universe) have been unsuccessful. Here we identify [C II]. = 157.74 mu m emission from two sources that we selected as high-redshift candidates on the basis of near-infrared photometry; we confirm that these sources are two galaxies at redshifts of z = 6.8540 +/- 0.0003 and z = 6.8076 +/- 0.0002. Notably, the luminosity of the [C II] line from these galaxies is higher than that found previously in star-forming galaxies with redshifts greater than 6.5. The luminous and extended [C II] lines reveal clear velocity gradients that, if interpreted as rotation, would indicate that these galaxies have similar dynamic properties to the turbulent yet rotation-dominated disks that have been observed in Ha-emitting galaxies two billion years later, at 'cosmic noon'.