  We present the results of the first, deep ALMA imaging covering the full 4.5
sq arcmin of the Hubble Ultra Deep Field (HUDF) as previously imaged with
WFC3/IR on HST. Using a mosaic of 45 pointings, we have obtained a homogeneous
1.3mm image of the HUDF, achieving an rms sensitivity of 35 microJy, at a
resolution of 0.7 arcsec. From an initial list of ~50 >3.5sigma peaks, a
rigorous analysis confirms 16 sources with flux densities S(1.3) > 120 microJy.
All of these have secure galaxy counterparts with robust redshifts (<z> =
2.15), and 12 are also detected at 6GHz in new deep JVLA imaging. Due to the
wealth of supporting data in this unique field, the physical properties of the
ALMA sources are well constrained, including their stellar masses (M*) and
UV+FIR star-formation rates (SFR). Our results show that stellar mass is the
best predictor of SFR in the high-z Universe; indeed at z > 2 our ALMA sample
contains 7 of the 9 galaxies in the HUDF with M* > 2 x 10^10 Msun and we detect
only one galaxy at z > 3.5, reflecting the rapid drop-off of high-mass galaxies
with increasing redshift. The detections, coupled with stacking, allow us to
probe the redshift/mass distribution of the 1.3-mm background down to S(1.3) ~
10 micro-Jy. We find strong evidence for a steep `main sequence' for
star-forming galaxies at z ~ 2, with SFR \propto M* and a mean specific SFR =
2.2 /Gyr. Moreover, we find that ~85% of total star formation at z ~ 2 is
enshrouded in dust, with ~65% of all star formation at this epoch occurring in
high-mass galaxies (M* > 2 x 10^10 Msun), for which the average
obscured:unobscured SF ratio is ~200. Finally, we combine our new ALMA results
with the existing HST data to revisit the cosmic evolution of star-formation
rate density; we find that this peaks at z ~ 2.5, and that the star-forming
Universe transits from primarily unobscured to primarily obscured thereafter at
z ~ 4.
