  The spin relaxation dynamics of Ce$^{3+}$ ions in heavily cerium-doped YAG
crystals is studied using pulse-electron paramagnetic resonance and
time-resolved pump-probe Faraday rotation. Both techniques address the 4$f$
ground state, while pump-probe Faraday rotation provides also access to the
excited 5$d$ state. We measure a millisecond spin-lattice relaxation time
$T_1$, a microsecond spin coherence time $T_2$ and a $\sim 10$ ns inhomogeneous
spin dephasing time $T_2^*$ for the Ce$^{3+}$ ground state at low temperatures.
The spin-lattice relaxation of Ce$^{3+}$ ions is due to modified Raman
processes involving the optical phonon mode at 125 cm$^{-1}$. The relaxation at
higher temperature goes through a first excited level of the $^{2}$F$_{5/2}$
term at about $\hbar \omega \approx 228$ cm$^{-1}$. Effects provided by the
hyperfine interaction of the Ce$^{3+}$ with the $^{27}$Al nuclei are observed.
