We present maps, source lists, and number counts from the largest, unbiassed, extragalactic sub-mm survey so far undertaken with the SCUBA camera on the JCMT. Our maps cover 260 sq. arcmin, to a noise level S(850)=2.5 mJy/beam. We have reduced the data using both SURF, and our own pipeline which produces zero-footprint maps and noise images. The uncorrelated noise maps produced by the latter approach have allowed application of a maximum-likelihood method to measure the statistical significance of each peak, leading to properly quantified flux-density errors for all potential sources. We detect 19 sources with S/N > 4, 38 with S/N > 3.5, and 72 with S/N > 3. To assess completeness and the impact of source confusion we have applied our source extraction algorithm to a series of simulated images. The result is a new estimate of the sub-mm source counts in the flux-density range S(850)=5-15mJy, which we compare with other estimates, and with model predictions. Our estimate of the cumulative source count at S(850) > 8 mJy is 320 (+80,-100) per square degree. Assuming that the majority of sources have z > 1.5, the co-moving number density of high-z galaxies forming stars at a rate >1000 solar masses per year is 10^-5 per Mpc^3, with only a weak dependence on the precise redshift distribution. This number density corresponds to that of massive ellipticals with L > 3-4 L* at low redshift and is also the same as the co-moving number density of comparably massive, passively-evolving objects in the redshift band 1<z<2 inferred from recent surveys of extremely red objects. Thus the bright sub-mm sources can plausibly account for the formation of all present-day massive ellipticals. Improved z constraints, and a proper measurement of sub-mm clustering can refine or refute this picture.