The autocorrelation function of a given process is related to its spectral density by the Wiener-Khintchine theorem, and both expressions contain the same information. We report here a measurement of the current noise produced in a lipid bilayer membrane doped with hydrophobic anions of dipicrylamine. The results are in good agreement both with relaxation measurements on the same membrane and with an analysis of the spectral density of the current noise for this system which has been presented by other workers. Although measurement of the spectral density function is generally more complete for technical reasons, the autocorrelation function provides, for the case studied here, more physical insight into the underlying charge transport mechanism. We find that the measured autocorrelation function is negative at short, but nonzero, times. This is a consequence of the operative conductance mechanism in this case, which cannot carry current continuously in the same direction without compensatory reverse flow.