  We consider the spatially homogeneous Boltzmann equation for inelastic
hard-spheres (with constant restitution coefficient $\alpha \in (0,1)$) under
the thermalization induced by a host medium with a fixed Maxwellian
distribution. We prove uniqueness of the stationary solution (with given mass)
in the weakly inelastic regime; i.e., for any inelasticity parameter $\alpha
\in (\alpha_0,1)$, with some constructive $\alpha_0 \in [0, 1)$. Our analysis
is based on a perturbative argument which uses the knowledge of the stationary
solution in the elastic limit and quantitative estimates of the convergence of
stationary solutions as the inelasticity parameter goes to 1. In order to
achieve this we give an accurate spectral analysis of the associated linearized
collision operator in the elastic limit. Several qualitative properties of this
unique steady state $F_\alpha$ are also derived; in particular, we prove that
$F_\alpha$ is bounded from above and from below by two explicit universal (i.e.
independent of $\alpha$) Maxwellian distributions.
