Price's Law states that linear perturbations of a Schwarzschild black hole, depending on initial conditions, fall off as $t^{-2\ell-3}$ or $t^{-2\ell-2}$ for $t \to \infty$ where $\ell$ is the angular momentum. We give a proof of such $\ell$--dependent decay rates in the form of weighted $L^1$ to $L^\infty$ bounds for solutions of the Regge--Wheeler equation. The proof is based on an integral representation of the solution which follows from self--adjoint spectral theory. We apply two different perturbative arguments in order to construct the corresponding spectral measure and the decay bounds are obtained by appropriate oscillatory integral estimates.