We map the dust distribution in the central 180" (~680pc) region of the M31 bulge, based on HST WFC3 and ACS observations in ten bands from near-ultraviolet (2700{AA}) to near-infrared (1.5-micron). This large wavelength coverage gives us great leverage to detect not only dense dusty clumps, but also diffuse dusty molecular gas. We fit a pixel-by-pixel spectral energy distributions to construct a high-dynamic-range extinction map with unparalleled angular resolution (~0.5", i.e., ~2pc) and sensitivity (the extinction uncertainty, delta A_V_~0.05). In particular, the data allow to directly fit the fractions of starlight obscured by individual dusty clumps, and hence their radial distances in the bulge. Most of these clumps seem to be located in a thin plane, which is tilted with respect to the M31 disk and appears face-on. We convert the extinction map into a dust mass surface density map and compare it with that derived from the dust emission as observed by Herschel. The dust masses in these two maps are consistent with each other, except in the low-extinction regions, where the mass inferred from the extinction tends to be underestimated. Further, we use simulations to show that our method can be used to measure the masses of dusty clumps in Virgo cluster early-type galaxies to an accuracy within a factor of ~2.