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 ( = 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.