  We report on the demonstration and characterization of a silicon optical
resonator for laser frequency stabilization, operating in the deep cryogenic
regime at temperatures as low as 1.5 K. Robust operation was achieved, with
absolute frequency drift less than 20 Hz over 1 hour. This stability allowed
sensitive measurements of the resonator thermal expansion coefficient
($\alpha$). We found $\alpha=4.6\times10^{-13}$ ${\rm K^{-1}}$ at 1.6 K. At
16.8 K $\alpha$ vanishes, with a derivative equal to $-6\times10^{-10}$ ${\rm
K}^{-2}$. The temperature of the resonator was stabilized to a level below 10
$\mu$K for averaging times longer than 20 s. The sensitivity of the resonator
frequency to a variation of the laser power was also studied. The corresponding
sensitivities and the expected Brownian noise indicate that this system should
enable frequency stabilization of lasers at the low-$10^{-17}$ level.
