  The surface density of submillimeter (sub-mm) galaxies as a function of flux,
usually termed the source number counts, constrains models of the evolution of
the density and luminosity of starburst galaxies. At the faint end of the
distribution, direct detection and counting of galaxies are not possible.
However, gravitational lensing by clusters of galaxies allows detection of
sources which would otherwise be too dim to study. We have used the largest
catalog of sub-mm-selected sources along the line of sight to galaxy clusters
to estimate the faint end of the 850 micron number counts; integrating to S =
0.10 mJy the equivalent flux density at 850 microns is v I_{v} = 0.24 +/- 0.03
nW/m^2/sr. This provides a lower limit to the extragalactic far-infrared
background and is consistent with direct estimates of the full intensity from
the FIRAS. The results presented here can help to guide strategies for upcoming
surveys carried out with single dish sub-mm instruments.
