We describe a method for measuring the frequency fluctuation spectrum of a laser oscillator, especially the weak noise contributions in the wings of the spectrum, and apply this method to confirm the existence of large excess quantum frequency fluctuations in a laser oscillator using an unstable optical resonator. Our measurement apparatus uses the Pound-Drever technique, which employs an RF phase modulator and a Fabry-Perot cavity to produce a sensitive high-speed frequency discrimination signal. We show that this signal can also be used to measure the quantum noise contributions to the frequency spectrum of a laser oscillator. Experimental measurements on a miniature diode-pumped Nd:YAG laser using a stable optical cavity closely match the predictions of the usual Schawlow-Townes theory, while the frequency fluctuations in a nearly identical laser employing an unstable optical resonator are approximately 1300 times larger. These much larger fluctuations arise in part from the larger output coupling and cavity bandwidth of the unstable cavity, but they also appear to confirm a predicted excess spontaneous emission factor (Petermann excess noise factor) of /spl ap/180 times arising from the nonorthogonal transverse mode properties of the unstable cavity. >