Frequency fluctuations of an optical frequency standard at 1.39 µm have been measured by means of a highly-sensitive optical frequency discriminator based on the fringe-side transmission of a high finesse optical resonator. Built on a Zerodur spacer, the optical resonator exhibits a finesse of 5500 and a cavity-mode width of about 120 kHz. The optical frequency standard consists of an extended-cavity diode laser that is tightly stabilized against the center of a sub-Doppler H 2 18 O line, this latter being detected by means of noise-immune cavity-enhanced optical heterodyne molecular spectroscopy. The emission linewidth has been carefully determined from the frequency-noise power spectral density by using a rather simple approximation, known as β-line approach, as well as the exact method based on the autocorrelation function of the laser light field. It turns out that the linewidth of the optical frequency standard amounts to about 7 kHz (full width at half maximum) for an observation time of 1 ms. Compared to the free-running laser, the measured width corresponds to a line narrowing by a factor of ~220.