  We prove exponential convergence in the energy norm of $hp$ finite element
discretizations for the integral fractional diffusion operator of order $2s\in
(0,2)$ subject to homogeneous Dirichlet boundary conditions in bounded
polygonal domains $\Omega\subset \mathbb{R}^2$. Key ingredient in the analysis
are the weighted analytic regularity from our previous work and meshes that
feature anisotropic geometric refinement towards $\partial\Omega$.
