Submillimetre and millimetre continuum observations of 10 IRAS-selected, radio-quiet quasars (RQQs) are presented. The first detections of three RQQs, PG 0050+124 (IZw 1), PG 0157+001 (Mrk 1014) and Mrk 376 have been made at 800 and 450 microns using the ^3^He bolometer UKT14 on the 15-m James Clerk Maxwell Telescope. These submillimetre data, together with existing 1.3-mm observations, demonstrate that the measured submillimetre spectral indices, (a_sm_) = 3.75+/-0.48, significantly exceed the critical theoretical limit of a_sm_ = 2.5 predicted for the self- absorption of synchrotron emission. This result is independent of any contributions to the 100-micron IRAS fluxes from cirrus emission in the host galaxies, extended circumnuclear star formation and FIR emission from companion or confusing sources. All current non-thermal models are rejected in favour of the alternative explanation that the FIR luminosity is dominated by thermal emission from warm (45-60 K) dust grains. The submillimetre optical depth and source-size for the thermal emission cannot yet be constrained by these data and, as a result, no discrimination can be made between dust heated by an extended (>1 kpc) starburst region or a central compact luminosity source. The high gas masses (>10^10^ M_sun_) in RQQs inferred from the submillimetre continuum observations are in agreement with the H_2_ masses determined from CO measurements. Alternatively, the results show that the M_H_2__/M_dust_ ratio measured in RQQs (~370+/-150) is consistent with that measured in spiral galaxies and ultraluminous IRAS galaxies.