  We investigate the fractional diffusion approximation of a kinetic equation
set in a bounded interval with diffusive reflection conditions at the boundary.
In an appropriate singular limit corresponding to small Knudsen number and long
time asymptotic, we show that the asymptotic density function is the {\it
unique solution} of a fractional diffusion equation with Neumann boundary
condition. This analysis completes a previous work by the same authors in which
a limiting fractional diffusion equation was identified on the half-space, but
the uniqueness of the solution (which is necessary to prove the convergence of
the whole sequence) could not be established.
