Observations of Cygnus X-3 at 2.0 and 1.1 mm with the JCMT, partially simultaneous with radio flux measurements at 2 cm and 18 cm, are presented. These are the first known observations of Cyg X-3 at millimetre wavelengths whilst the source was in a quiet radio state. The observed millimetre fluxes are several times greater than those expected for the synchrotron tail of emission extrapolated from the radio, and the 2-mm flux exceeds that at 1.1 mm, suggesting a peak in the region of a few mm. The observations are interpreted using an extension of the model of Marti, Paredes & Estalella to repeated short injections of relativistic electrons in a region in the Cyg X-3 stellar wind which is optically thin to millimetre-wave photons, but optically thick at centimetre wavelengths. Centimetre-wavelength emission is not observed until the ejecta have travelled further out in the wind. A MERLIN radio map at 18 cm reveals the same elliptical asymmetry in the scattering medium around Cyg X-3 as found by Wilkinson, Narayan & Spencer.