We prove the emergence of stable fluctuations for reaction-diffusion in random environment with Weibull tails. This completes our work around the quenched to annealed transition phenomenon in this context of reaction diffusion. In [9], we had already considered the model treated here and had studied fully the regimes where the law of large numbers is satisfied and where the fluctuations are Gaussian, but we had left open the regime of stable fluctuations. Our work is based on a spectral approach centered on the classical theory of rank-one perturbations. It illustrates the gradual emergence of the role of the higher peaks of the environments. This approach also allows us to give the delicate exact asymptotics of the normalizing constants needed in the stable limit law.