  We report the first counts of faint submillimetre galaxies (SMG) in the
870-um band derived from arcsecond resolution observations with the Atacama
Large Millimeter Array (ALMA). We have used ALMA to map a sample of 122
870-um-selected submillimetre sources drawn from the (0.5x0.5)deg^2 LABOCA
Extended Chandra Deep Field South Submillimetre Survey (LESS). These ALMA maps
have an average depth of sigma(870um)~0.4mJy, some ~3x deeper than the original
LABOCA survey and critically the angular resolution is more than an order of
magnitude higher, FWHM of ~1.5" compared to ~19" for the LABOCA discovery map.
This combination of sensitivity and resolution allows us to precisely pin-point
the SMGs contributing to the submillimetre sources from the LABOCA map, free
from the effects of confusion. We show that our ALMA-derived SMG counts broadly
agree with the submillimetre source counts from previous, lower-resolution
single-dish surveys, demonstrating that the bulk of the submillimetre sources
are not caused by blending of unresolved SMGs. The difficulty which
well-constrained theoretical models have in reproducing the high-surface
densities of SMGs, thus remains. However, our observations do show that all of
the very brightest sources in the LESS sample, S(870um)>12mJy, comprise
emission from multiple, fainter SMGs, each with 870-um fluxes of <9mJy. This
implies a natural limit to the star-formation rate in SMGs of <10^3 M_Sun/yr,
which in turn suggests that the space densities of z>1 galaxies with gas masses
in excess of ~5x10^10 M_Sun is <10^-5 Mpc^-3. We also discuss the influence of
this blending on the identification and characterisation of the SMG
counterparts to these bright submillimetre sources and suggest that it may be
responsible for previous claims that they lie at higher redshifts than fainter
SMGs.
