Using the SCUBA bolometer array on the JCMT, we have carried out a submillimetre survey of Broad Absorption Line quasars (BALQs). The sample has been chosen to match, in redshift and optical luminosity, an existing benchmark 850um sample of radio-quiet quasars, allowing a direct comparison of the submm properties of BAL quasars relative to the parent radio-quiet population. We reach a submm limit 1.5mJy at 850um, allowing a more rigorous measure of the submm properties of BAL quasars than previous studies. Our submm photometry complements extensive observations at other wavelengths, in particular X-rays with Chandra and mid-infrared with Spitzer. To compare the 850um flux distribution of BALQs with that of the non-BAL quasar benchmark sample, we employ a suite of statistical methods, including survival analysis and a novel Bayesian derivation of the underlying flux distribution. Although there are no strong grounds for rejecting the null hypothesis that BALQs on the whole have the same submm properties as non-BAL quasars, we do find tentative evidence (1-4 percent significance from a K-S test and survival analysis) for a dependence of submm flux on the equivalent width of the characteristic CIV broad absorption line. If this effect is real - submm activity is linked to the absorption strength of the outflow - it has implications either for the evolution of AGN and their connection with star formation in their host galaxies, or for unification models of AGN.