  We study one-magnon excitations in a random ferromagnetic Heisenberg chain
with long-range correlations in the coupling constant distribution. By
employing an exact diagonalization procedure, we compute the localization
length of all one-magnon states within the band of allowed energies $E$. The
random distribution of coupling constants was assumed to have a power spectrum
decaying as $S(k)\propto 1/k^{\alpha}$. We found that for $\alpha < 1$,
one-magnon excitations remain exponentially localized with the localization
length $\xi$ diverging as 1/E. For $\alpha = 1$ a faster divergence of $\xi$ is
obtained. For any $\alpha > 1$, a phase of delocalized magnons emerges at the
bottom of the band. We characterize the scaling behavior of the localization
length on all regimes and relate it with the scaling properties of the
long-range correlated exchange coupling distribution.
