Low noise and high repetition rate optical frequency combs are desirable for many applications from timekeeping to precision spectroscopy. For example, gigahertz repetition rate sources greatly increase the acquisition speed of spectra in a dual-comb modality when compared to lower repetition rate sources, while still maintaining sufficient instantaneous resolution to resolve ro-vibrational signatures from molecules in a variety of conditions. In this paper, we present the stabilization and characterization of a turnkey commercial 1~GHz mode-locked laser that operates at telecom wavelengths (1.56 $mu$m). Fiber amplification and spectral broadening result in the high signal-to-noise ratio detection and stabilization of $textit{f}_{textit{ceo}}$ with 438 mrad of residual phase noise (integrated from 10$^2$ to 10$^7$ Hz). Simultaneously, we stabilize the beatnote between the nearest comb mode and a cavity stabilized continuous-wave laser at 1.55 $mu$m with 41 mrad of residual phase noise (integrated from 10$^2$ to 10$^7$ Hz). This robust, self-referenced comb system is built with off-the-shelf polarization-maintaining fiber components and will be useful for a wide range of low noise frequency comb applications that benefit from the increased repetition rate.
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