  We have exploited ALMA calibration observations to carry out a novel, wide
and deep submm survey, ALMACAL. These calibration data comprise a large number
of observations of calibrator fields in a variety of frequency bands and array
configurations. Gathering together data acquired during multiple visits to many
ALMA calibrators, it is possible to reach noise levels which allow the
detection of faint dusty, star-forming galaxies (DSFGs) over a significant
area. In this paper we outline our survey strategy and report the first
results. We have analysed data for 69 calibrators, reaching depths of $\sim 25
\, {\rm \mu Jy \, beam^{-1}}$ at sub-arcsec resolution. Adopting a conservative
approach based on $\geq 5 \sigma$ detections, we have found eight and 11 DSFGs
in ALMA bands 6 and 7, respectively, with flux densities $S_{\rm 1.2 mm} \geq
0.2 \, {\rm mJy}$. The faintest galaxies would have been missed by even the
deepest \emph{Herschel} surveys. Our cumulative number counts have been
determined independently at 870 $\mu$m and 1.2 mm, from a sparse sampling of
the astronomical sky, and are thus relatively free of cosmic variance. The
counts are lower than reported previously by a factor of at least $2\times$.
Future analyses will yield large, secure samples of DSFGs, with redshifts
determined via detection of submm spectral lines. Uniquely, our strategy then
allows morphological studies of very faint DSFGs - representative of more
normal star-forming galaxies than conventional submm galaxies (SMGs) - in
fields where self-calibration is feasible, yielding milliarcsecond spatial
resolution.
