  We analyze four epochs of HST imaging over 18 years for the Draco dwarf
spheroidal galaxy. We measure precise proper motions (PMs) for hundreds of
stars and combine these with existing line-of-sight (LOS) velocities. This
provides the first radially-resolved 3D velocity dispersion profiles for any
dwarf galaxy. These constrain the intrinsic velocity anisotropy and resolve the
mass-anisotropy degeneracy. We solve the Jeans equations in oblate axisymmetric
geometry to infer the mass profile. We find the velocity dispersion to be
radially anisotropic along the symmetry axis and tangentially anisotropic in
the equatorial plane, with a globally-averaged value $\overline{\beta_{\mathrm
B}}=-0.20^{+ 0.28}_{- 0.53}$, (where $1 - \beta_{\mathrm B} \equiv \langle
v_{\mathrm{ tan}}^2 \rangle / \langle v_{\mathrm{ rad}}^2 \rangle$ in 3D). The
logarithmic dark matter (DM) density slope over the observed radial range,
$\Gamma_{\mathrm{ dark}}$, is $-0.83^{+ 0.32}_{- 0.37}$, consistent with the
inner cusp predicted in $\Lambda$CDM cosmology. As expected given Draco's low
mass and ancient star formation history, it does not appear to have been
dissolved by baryonic processes. We rule out cores larger than 487, 717, 942 pc
at respective 1-, 2-, 3-$\sigma$ confidence, thus imposing important
constraints on the self-interacting DM cross-section. Spherical models yield
biased estimates for both the velocity anisotropy and the inferred slope. The
circular velocity at our outermost data point (900 pc) is $24.19^{+ 6.31}_{-
2.97} \ \mathrm{km~s^{-1}}s$. We infer a dynamical distance of $75.37^{+
4.73}_{- 4.00}$ kpc, and show that Draco has a modest LOS rotation, with
$\left<v / \sigma \right> = 0.22 \pm 0.09$. Our results provide a new stringent
test of the so-called `cusp-core' problem that can be readily extended to other
dwarfs.
