Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three

For each fixed dimension $d\ge3$ and each fixed sufficiently small positive disorder strength, we prove that the Anderson operator on $\mathbb Z^d$ with independent uniform site potentials almost surely has purely absolutely continuous spectrum with nonzero weight on an open energy interval. The interval may depend on $d$ but is independent of the disorder strength. This settles the purely absolutely continuous energy-range question in Simon's Problem 1 for this model.