We present millimetre photometry and submillimetre imaging of the central core and two hotspots in the radio lobes of the galaxy Cygnus A. For both hotspots and the central core, the synchrotron spectrum continues smoothly from the radio to a frequency of 677 GHz. The spectral index of the hotspots is constant over our frequency range, with a spectral index of α ≈ −1.0 (Sν ∝ να), which is steeper than at lower frequencies and represents the emission from an aged population of electrons. The core is significantly flatter, with α = −0.6 ± 0.1, suggestive of an injected spectrum with no ageing, but some evidence for steepening exists at our highest observing frequency. Although IRAS data suggest the presence of dust in Cygnus A, our 450-μm data show no evidence of cold dust, therefore the dust component must have a temperature lying between 85 and 37 K, corresponding to dust masses of 1.4 × 106 and 1.0 × 108 M⊙ respectively.