We present a sensitive 870 micron survey of the Extended Chandra Deep Field South (ECDFS) using LABOCA on the APEX telescope. The LABOCA ECDFS Submillimetre Survey (LESS) covers the full 30' x 30' field size of the ECDFS and has a uniform noise level of 1.2 mJy/beam. LESS is thus the largest contiguous deep submillimetre survey undertaken to date. The noise properties of our map show clear evidence that we are beginning to be affected by confusion noise. We present a catalog of 126 SMGs detected with a significance level above 3.7 sigma. The ECDFS exhibits a deficit of bright SMGs relative to previously studied blank fields but not of normal star-forming galaxies that dominate the extragalactic background light (EBL). This is in line with the underdensities observed for optically defined high redshift source populations in the ECDFS (BzKs, DRGs,optically bright AGN and massive K-band selected galaxies). The differential source counts in the full field are well described by a power law with a slope of alpha=-3.2, comparable to the results from other fields. We show that the shape of the source counts is not uniform across the field. The integrated 870 micron flux densities of our source-count models account for >65% of the estimated EBL from COBE measurements. We have investigated the clustering of SMGs in the ECDFS by means of a two-point correlation function and find evidence for strong clustering on angular scales 1. Assuming a power law dependence for the correlation function and a typical redshift distribution for the SMGs we derive a spatial correlation length of r_0=13+/-6 h^-1 Mpc.