  We explore the potential of direct spectroscopy of ultra-narrow optical
transitions of atoms localized in an optical cavity. In contrast to
stabilization against a reference cavity, which is the approach currently used
for the most highly stabilized lasers, stabilization against an atomic
transition does not suffer from Brownian thermal noise. Spectroscopy of
ultra-narrow optical transitions in a cavity operates in a very highly
saturated regime in which non-linear effects such as bistability play an
important role. From the universal behavior of the Jaynes-Cummings model with
dissipation, we derive the fundamental limits for laser stabilization using
direct spectroscopy of ultra-narrow atomic lines. We find that with current
lattice clock experiments, laser linewidths of about 1 mHz can be achieved in
principle, and the ultimate limitations of this technique are at the 1 $\mu$ Hz
level.
