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We perform hole burning with a low drift stabilized laser within the zerophonon line of the 4f-5d transition in Ce$^{3+}$:Y$_2$SiO$_5$ at 2K. Thenarrowest spectral holes appear for small applied magnetic fields and are$6\pm4$ MHz wide (FWHM). This puts an upper bound on the homogeneous linewidthof the transition to $3\pm2$ MHz, which is close to lifetime limited. The spinlevel relaxation time is measured to $72\pm21$ ms with a magnetic field of 10mT.  A slow permanent hole burning mechanism is observed. If the excitationfrequency is not changed the fluorescence intensity is reduced by more than50$\%$ after a couple of minutes of continuous excitation. The spectral holecreated 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.

High resolution transient and permanent spectral hole burning in Ce$^{3+}$:Y$_2$SiO$_5$ at liquid helium temperatures