  We present and analyze a method of laser frequency stabilization via
steady-state patterns of spectral holes in Eu$^{3+}$:Y$_2$SiO$_5$. Three
regions of spectral holes are created, spaced in frequency by the ground state
hyperfine splittings of $~^{151}$Eu$^{3+}$. The absorption pattern is shown not
to degrade after days of laser frequency stabilization. An optical frequency
comparison of a laser locked to such a steady-state spectral-hole pattern with
an independent cavity-stabilized laser and a Yb optical lattice clock
demonstrates a spectral-hole fractional frequency instability of
$1.0\times10^{-15}~ \tau^{-\frac{1}{2}}$ that averages to
$8.5^{+4.8}_{-1.8}\times10^{-17}$ at $\tau = 73$ s. Residual amplitude
modulation at the frequency of the RF drive applied to the fiber-coupled
electro-optic modulator is reduced to less than $1\times10^{-6}$ fractional
amplitude modulation at $\tau>$ 1 s by an active servo. The contribution of
residual amplitude modulation to the laser frequency instability is further
reduced by digital division of the transmission and incident photodetector
signals to less than $1\times10^{-16}$ at $\tau>$ 1 s.
