In some models of the early Universe, radiation is reprocessed into the submillimetre and far-infrared by high-redshift dust, without violating COBE limits on the CBR spectrum, but producing secondary anisotropies which should be detectable with ground-based submillimetre telescopes. We describe an attempt to measure these anisotropies at 800 mum using the JCMT. A careful analysis to reduce position-dependent systematics was carried out and we show that, for this experiment, chopping the telescope beam in azimuth rather than RA produces lower systematics. Bayesian likelihood analysis is then used to set an upper limit on CBR fluctuations of DELTAT/T less-than-or-similar-to 1.46 X 10(-3) on a coherence scale of 17 arcsec. Data from a similar set of observations made by Kreysa & Chini in 1988 with the IRAM 30-m telescope at 1300 mum are reanalysed in the same way to enable a comparison to be made. The results are used to set limits on the generation of cosmic backgrounds from primeval dust.