  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.
