We make use of deep 1.2 mm continuum observations (12.7 μJy beam-1 rms) of a 1 arcmin2 region in the Hubble Ultra Deep Field to probe dust-enshrouded star formation from 330 Lyman-break galaxies spanning the redshift range z = 2-10 (to ∼2-3 M o yr-1 at 1σ over the entire range). Given the depth and area of ASPECS, we would expect to tentatively detect 35 galaxies, extrapolating the Meurer z ∼ 0 IRX-β relation to z ≥ 2 (assuming dust temperature T d ∼ 35 K). However, only six tentative detections are found at z 2 in ASPECS, with just three at >3σ. Subdividing our z = 2-10 galaxy samples according to stellar mass, UV luminosity, and UV-continuum slope and stacking the results, we find a significant detection only in the most massive (>109.75 M o) subsample, with an infrared excess (IRX = L IR/L UV) consistent with previous z ∼ 2 results. However, the infrared excess we measure from our large selection of sub-L (<109.75Mo) galaxies is ±0.34 (bootstrap and formal uncertainties) and ±0.18 at z = 2-3 and z = 4-10, respectively, lying below even an IRX-β relation for the Small Magellanic Cloud (95% confidence). These results demonstrate the relevance of stellar mass for predicting the IR luminosity of z 2 galaxies. We find that the evolution of the IRX-stellar mass relationship depends on the evolution of the dust temperature. If the dust temperature increases monotonically with redshift () such that T d ∼ 44-50 K at z ≥ 4, current results are suggestive of little evolution in this relationship to z ∼ 6. We use these results to revisit recent estimates of the z ≥ 3 star formation rate density.