Correcting Type Ia Supernova brightnesses for extinction by dust has proven to be a vexing problem. Here we study the dust foreground to the highly reddened SN 2012cu, which is projected onto a dust lane in the galaxy NGC 4772. The analysis is based on multi-epoch, spectrophotometric observations spanning from 3300-9200 Å, obtained by the Nearby Supernova Factory. Phase-matched comparison of the spectroscopically twinned SN 2012cu and SN 2011fe across 10 epochs results in the best-fit color excess of (E (B - V), RMS) = (1.00, 0.03) and total-to-selective extinction ratio of (RV, RMS) = (2.95, 0.08) toward SN 2012cu within its host galaxy. We further identify several diffuse interstellar bands and compare the 5780 Å band with the dust-to-band ratio for the Milky Way (MW). Overall, we find the foreground dust-extinction properties for SN 2012cu to be consistent with those of the MW. Furthermore, we find no evidence for significant time variation in any of these extinction tracers. We also compare the dust extinction curve models of Cardelli et al., ODonnell, and Fitzpatrick, and find the predictions of Fitzpatrick fit SN 2012cu the best. Finally, the distance to NGC4772, the host of SN 2012cu, at a redshift of z = 0.0035, often assigned to the Virgo Southern Extension, is determined to be 16.6 ± 1.1 Mpc. We compare this result with distance measurements in the literature.