We present the results of the first, deep Atacama Large Millimeter Array ( ALMA) imaging covering the full similar or equal to 4.5 arcmin(2) of the Hubble Ultra Deep Field ( HUDF) imaged with Wide Field Camera 3/IR on HST. Using a 45-pointing mosaic, we have obtained a homogeneous 1.3-mm image reaching sigma 1.3 similar or equal to 35 mu Jy, at a resolution of similar or equal to 0.7 arcsec. From an initial list of similar or equal to 50 > 3.5 sigma peaks, a rigorous analysis confirms 16 sources with S-1.3 > 120 mu Jy. All of these have secure galaxy counterparts with robust redshifts (< z > = 2.15). Due to the unparalleled supporting data, 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-redshift Universe; indeed at z = 2 our ALMA sample contains seven of the nine galaxies in the HUDF withM(*) = 2 x 10(10)M circle dot, 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 S1.3 similar or equal to 10 mu Jy. We find strong evidence for a steep star-forming `main sequence' at z similar or equal to 2, with SFR. M* and a mean specific SFR similar or equal to 2.2 Gyr(-1). Moreover, we find that similar or equal to 85 per cent of total star formation at z similar or equal to 2 is enshrouded in dust, with similar or equal to 65 per cent of all star formation at this epoch occurring in high-mass galaxies ( M-* > 2 x 10(10)M circle dot), for which the average obscured: unobscured SF ratio is similar or equal to 200. Finally, we revisit the cosmic evolution of SFR density; we find this peaks at z similar or equal to 2.5, and that the star-forming Universe transits from primarily unobscured to primarily obscured at z similar or equal to 4.