  We present deep 850 micron maps of three massive lensing clusters, A370,
A851, and A2390, with well-constrained mass models. Our cluster exposure times
are more than 2 to 5 times longer than any other published cluster field
observations. We catalog the sources and determine the submillimeter number
counts. The counts are best determined in the 0.3 to 2 mJy range where the
areas are large enough to provide a significant sample. At 0.3 mJy the
cumulative counts are 3.3 (1.3,6.3) 10^4 per square degree, where the upper and
lower bounds in the brackets are the 90% confidence range. The surface density
at these faint count limits enters the realm of significant overlap with other
galaxy populations.The corresponding percentage of the extragalactic background
light (EBL) residing in this flux range is about 45-65%, depending on the EBL
measurement used. Given that 20-30% of the EBL is resolved at flux densities
between 2 and 10 mJy, most of the submillimeter EBL is arising in sources above
0.3 mJy. We also performed a noise analysis to obtain an independent estimate
of the counts. The upper bounds on the counts determined from the noise
analysis closely match the upper limits obtained from the direct counts. The
differential counts from this and other surveys can reasonably be described by
the parameterization n(S)=3 10^4/(0.7 + S^3) per square degree per mJy with S
in mJy, which also integrates to match the EBL.
