Models and simulations(1-4) of the epoch of reionization predict that spectra of the 21-centimetre transition of atomic hydrogen will show a clear fluctuation peak, at a redshift and scale, respectively, that mark the central stage of reionization and the characteristic size of ionized bubbles. This is based on the assumption(5-7) that the cosmic gas was heated by stellar remnants-particularly X-ray binaries-to temperatures well above the cosmic microwave background at that time (about 30 kelvin). Here we show instead that the hard spectra (that is, spectra with more high-energy photons than low-energy photons) of X-ray binaries(8,9) make such heating ineffective, resulting in a delayed and spatially uniform heating that modifies the 21-centimetre signature of reionization. Rather than looking for a simple rise and fall of the large-scale fluctuations (peaking at several millikelvin), we must expect a more complex signal also featuring a distinct minimum (at less than a millikelvin) that marks the rise of the cosmic mean gas temperature above the microwave background. Observing this signal, possibly with radio telescopes in operation today, will demonstrate the presence of a cosmic background of hard X-rays at that early time.