Synchronously pumped optical parametric oscillators (OPOs) are important tools for frequency comb (FC) generation in the mid-IR spectral range, where few suitable laser gain materials exist. For degenerate OPOs, self-phase-locking to the pump FC has been demonstrated. Here, we present a phase noise study of the carrier envelope offset frequency, revealing a -6 dB reduction compared to the pump FC over a wide Fourier frequency range. These results demonstrate that a degenerate OPO can be an ideal coherent frequency divider without any excess noise.