  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.
