  Length changes due to thermo-mechanical noise originating from, for example,
Brownian motion are a key limiting factor of present day state-of-the-art laser
frequency stabilization using Fabry-P\'erot cavities. We present a
laser-frequency stabilization concept using an optical cavity with a strong
slow-light effect to reduce the impact of cavity length changes on the
frequency stability. The resulting noise-reduction factor is proportional to
the ratio between the light phase and group velocities in the highly dispersive
cavity spacer. We experimentally demonstrate a proof-of-principle
implementation of this laser-frequency stabilization technique using a
rare-earth doped crystalline cavity spacer in conjunction with semi-permanent
spectral tailoring to achieve precise control of the dispersive properties of
the cavity. Compared to the same setup in the absence of the slow-light effect
a reduction in frequency sensitivity of four orders of magnitude was achieved.
