  Robust control and stabilization of optical frequency combs enables an
extraordinary range of scientific and technological applications, including
frequency metrology at extreme levels of precision, novel spectroscopy of
quantum gases and of molecules from visible wavelengths to the far infrared,
searches for exoplanets, and photonic waveform synthesis. Here we report on the
stabilization of a microresonator-based optical comb (microcomb) by way of
mechanical actuation. This represents an important step in the development of
microcomb technology, which offers a pathway toward fully-integrated comb
systems. Residual fluctuations of our 32.6 GHz microcomb line spacing reach a
record stability level of $5\times10^{-15}$ for 1 s averaging, thereby
highlighting the potential of microcombs to support modern optical frequency
standards. Furthermore, measurements of the line spacing with respect to an
independent frequency reference reveal the effective stabilization of different
spectral slices of the comb with a $<$0.5 mHz variation among 140 comb lines
spanning 4.5 THz. These experiments were performed with newly-developed
microrod resonators, which were fabricated using a CO$_2$-laser-machining
technique.
