The persistent current in an ensemble of normal-metal rings shows Gaussian distributed sample-to-sample fluctuations with non-Gaussian corrections, which are precursors of the transition into the Anderson localized regime. We here report a calculation of the leading non-Gaussian correction to the current autocorrelation function, which is of third-order in the current. Although the third-order correlation function is small, inversely proportional to the dimensionless conductance g of the ring, the mere fact that it is nonzero is remarkable, since it is an odd moment of the current distribution.