  We prove stability estimates for the spatially discrete, Galerkin solution of
a fractional Fokker-Planck equation, improving on previous results in several
respects. Our main goal is to establish that the stability constants are
bounded uniformly in the fractional diffusion exponent $\alpha\in(0,1]$. In
addition, we account for the presence of an inhomogeneous term and show a
stability estimate for the gradient of the Galerkin solution. As a by-product,
the proofs of error bounds for a standard finite element approximation are
simplified.
