We present a detailed study of the colours of late-type galaxy discs for ten of the EDisCS galaxy clusters with 0.5 less than or similar to z less than or similar to 0.8. Our cluster sample contains 172 spiral galaxies, and our control sample is composed of 96 field disc galaxies. We deconvolved their ground-based V and I images obtained with FORS2 at the VLT with initial spatial resolutions between 0.4 and 0.8 arcsec to achieve a final resolution of 0.1 arcsec with 0.05 arcsec pixels, which is close to the resolution of the ACS at the HST. After removing the central region of each galaxy to avoid pollution by the bulges, we measured the V - I colours of the discs. We find that 50% of cluster spiral galaxies have disc V - I colours redder by more than 1 sigma of the mean colours of their field counterparts. This is well above the 16% expected for a normal distribution centred on the field disc properties. The prominence of galaxies with red discs depends neither on the mass of their parent cluster nor on the distance of the galaxies to the cluster cores. Passive spiral galaxies constitute 20% of our sample. These systems are not abnormally dusty. They are are made of old stars and are located on the cluster red sequences. Another 24% of our sample is composed of galaxies that are still active and star forming, but less so than galaxies with similar morphologies in the field. These galaxies are naturally located in the blue sequence of their parent cluster colour-magnitude diagrams. The reddest of the discs in clusters must have stopped forming stars more than similar to 5 Gyr ago. Some of them are found among infalling galaxies, suggesting preprocessing. Our results confirm that galaxies are able to continue forming stars for some significant period of time after being accreted into clusters, and suggest that star formation can decline on seemingly long (1 to 5 Gyr) timescales.