  We perform hole burning with a low drift stabilized laser within the zero
phonon line of the 4f-5d transition in Ce$^{3+}$:Y$_2$SiO$_5$ at 2K. The
narrowest spectral holes appear for small applied magnetic fields and are
$6\pm4$ MHz wide (FWHM). This puts an upper bound on the homogeneous linewidth
of the transition to $3\pm2$ MHz, which is close to lifetime limited. The spin
level relaxation time is measured to $72\pm21$ ms with a magnetic field of 10
mT.
  A slow permanent hole burning mechanism is observed. If the excitation
frequency is not changed the fluorescence intensity is reduced by more than
50$\%$ after a couple of minutes of continuous excitation. The spectral hole
created by the permanent hole burning has a width in the tens of MHz range,
which indicates that a trapping mechanism occurs via the 5d-state.
