We have observed continuum emission at lambda = 850 microns over ~4 square degrees of the Ophiuchus star-forming cloud using SCUBA on the JCMT, producing a submillimetre continuum map twenty times larger than previous Ophiuchus surveys. Our sensitivity is 40 mJ/beam, a factor of ~2 less sensitive than earlier maps. Using an automated identification algorithm, we detect 100 candidate objects. Only two new objects are detected outside the boundary of previous maps, despite the much wider area surveyed. We compare the submillimetre continuum map with a map of visual extinction across the Ophiuchus cloud derived using a combination of 2MASS and R-band data. The total mass in submillimetre objects is ~ 50 Msun compared with ~ 2000 Msun in observed cloud mass estimated from the extinction. The submillimetre objects represent only 2.5% of the cloud mass. A clear association is seen between the locations of detected submillimetre objects and high visual extinction, with no objects detected at A_V<7 magnitudes. Using the extinction map, we estimate pressures within the cloud from P/k ~2x10^5 cm^-3 K in the less-extincted regions to P/k ~2x10^6 cm^-3 K at the cloud centre. Given our sensitivities, cold (T_d ~15K) clumps supported by thermal pressure, had they existed, should have been detected throughout the majority of the map. Such objects may not be present at low A_V because they may form only where A_V > 15, by some mechanism (e.g., loss of non-thermal support). Comment: 12 pages, 1 figure. Accepted by Astrophysical Journal Letters