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Determination of the mass density profiles of dwarf galaxies (and specifically whether there is a central core or cusp) provides a critical test of both the properties of dark matter (DM) and the physics of cosmological structure formation. The nearby classical dwarf spheroidal galaxies (dSphs) of the Milky Way yield some of the best dynamical constraints. While large line-of-sight velocity datasets exist (some thousand stars per galaxy) comma interpretation is hindered by the well-known mass vs. velocity-anisotropy degeneracy of stellar dynamics. This can be resolved with proper motion (PM) measurements that yield 3-D velocity information. This is beyond the reach of Gaia comma given the small velocity dispersions of dSphs and the absence of bright stars. HST is the only observatory that can advance this problem comma given its combination of photometric depth comma high spatial resolution comma and long time baselines. We propose to obtain HST imaging of five previously imaged fields in the nearby Draco and Sculptor dSphs comma to obtain high-accuracy PMs for thousands of stars in these galaxies. This provides a direct determination of their velocity anisotropy profiles comma and combined with dynamical models comma tightly constrains the slopes of their DM density profiles. The results will give unique constraints on both the nature of DM comma and the physical mechanisms that shape DM density profiles in galaxies. No comparable measurements exist to date. The proposed program therefore shows how HST can still be used comma after 30 years in orbit comma to tackle unanswered fundamental questions in astrophysics.