The low-frequency radio spectra of the hotspots within powerful radio galaxies can provide valuable information about the physical processes operating at the site of the jet termination. These processes are responsible for the dissipation of jet kinetic energy, particle acceleration, and magnetic-field generation. Here, we report new observations of the powerful radio galaxy Cygnus A using the Low Frequency Array (LOFAR) between 109 and 183 MHz, at an angular resolution of ∼3.5 arcsec. The radio emission of the lobes is found to have a complex spectral index distribution, with a spectral steepening found towards the centre of the source. For the first time, a turnover in the radio spectrum of the two main hotspots of Cygnus A has been directly observed. By combining our LOFAR imaging with data from the Very Large Array at higher frequencies, we show that the very rapid turnover in the hotspot spectra cannot be explained by a low-energy cut-off in the electron energy distribution, as has been previously suggested. Thermal (free–free) absorption or synchrotron self-absorption models are able to describe the low-frequency spectral shape of the hotspots; however, as with previous studies, we find that the implied model parameters are unlikely, and interpreting the spectra of the hotspots remains problematic.