Autocorrelations due to natural phase and amplitude fluctuations of the laser field were investigated in the threshold region of a 633-nm He-Ne single-mode laser by use of an' interferometric autodyne method. The measured time-dependent autocorrelation function shows deviations from gn exponential decay that can be described by a formal fine structure predicted by the theoretical Van der Pol laser model. As a consequence, the natural laser spectrum is not exactly Lorentz shaped. The natural linewidth shows the expected general reciprocal power dependence and an additional factor-of-2 narrowing above threshold, in full agreement with the theory.