We make use of sensitive (9.3 μJy beam-1 rms) 1.2 mm continuum observations from the Atacama Large Millimeter/submillimeter Array (ALMA) Spectroscopic Survey in the Hubble Ultra-Deep Field (ASPECS) large program to probe dust-enshrouded star formation from 1362 Lyman-break galaxies spanning the redshift range z = 1.5-10 (to ∼7-28 M o˙ yr-1 at 4σ over the entire range). We find that the fraction of ALMA-detected galaxies in our z = 1.5-10 samples increases steeply with stellar mass, with the detection fraction rising from 0% at 109.0 M o˙ to 85-18+9% at >1010 M o˙. Moreover, on stacking all 1253 low-mass (<109.25 M o˙) galaxies over the ASPECS footprint, we find a mean continuum flux of -0.1 0.4 μJy beam-1, implying a hard upper limit on the obscured star formation rate of <0.6 M o˙ yr-1 (4σ) in a typical low-mass galaxy. The correlation between the infrared excess (IRX) of UV-selected galaxies (L IR/L UV) and the UV-continuum slope is also seen in our ASPECS data and shows consistency with a Calzetti-like relation at > 109.5Ṁ and an SMC-like relation at lower masses. Using stellar mass and β measurements for z ∼ 2 galaxies over the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey, we derive a new empirical relation between β and stellar mass and then use this correlation to show that our IRX-β and IRX-stellar mass relations are consistent with each other. We then use these constraints to express the IRX as a bivariate function of β and stellar mass. Finally, we present updated estimates of star formation rate density determinations at z > 3, leveraging present improvements in the measured IRX and recent probes of ultraluminous far-IR galaxies at z > 2.