Summary: 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 that the solution to the associated initial-value problem converges exponentially fast towards the unique equilibrium solution. The proof combines a careful spectral analysis of the linearised semigroup as well as entropy estimates. The trend towards equilibrium holds in the weakly inelastic regime in which \(\alpha\) is close to 1, and the rate of convergence is explicit and depends solely on the spectral gap of the elastic linear collision operator.