  The frequency stability of a laser locked to an optical reference cavity is
fundamentally limited by thermal noise in the cavity length. These fluctuations
are linked to material dissipation, which depends both on the temperature of
the optical components and the material properties. Here, the design and
experimental characterization of a sapphire optical cavity operated at 10 K
with crystalline coatings at 1069 nm is presented. Theoretical estimates of the
thermo-mechanical noise indicate a thermal noise floor below
$\mathrm{4.5\times10^{-18}}$. Major technical noise contributions including
vibrations, temperature fluctuations, and residual amplitude modulation are
characterized in detail. The short-term performance is measured via a
three-cornered hat analysis with two other cavity-stabilized lasers, yielding a
noise floor of $1\times10^{-16}$. The long-term performance is measured against
an optical lattice clock, indicating cavity stability at the level of
$2\times10^{-15}$ for averaging times up to 10,000 s.
