This paper deals with noncoherent discrete-time detection of a narrow-band signal subject to slow and nonselective fading and embedded in correlated non-Gaussian noise modeled as a spherically invariant random process whose modulating random variable is continuous. At first, an asymptotic sufficient statistic for an arbitrary fading law is derived; then, the asymptotically optimum detector for Rayleigh-distributed amplitude fluctuations is synthesized. The detection structure implementation requires the knowledge, but for a scale factor, of the correlation function of the noise, but is independent of the distribution function of the modulating random variable. The performance of the asymptotically optimum detector synthesized for Rayleigh fading is assessed via computer simulations. The results show that the performance degradation with respect to the fully optimum performance is scarcely significant, even for low values of the sample size. Moreover, in highly non-Gaussian noise the proposed detector largely outperforms the fully optimum detector synthesized under the correlated Gaussian noise assumption.