  The spin-lattice relaxation time, $T_{1}$, of conduction electrons is
measured as a function of temperature and magnetic field in MgB$_2$. The method
is based on the detection of the $z$ component of the conduction electron
magnetization under electron spin resonance conditions with amplitude modulated
microwave excitation. Measurement of $T_{1}$ below $T_c$ at 0.32 T allows to
disentangle contributions from the two Fermi surfaces of MgB$_{2}$ as this
field restores normal state on the Fermi surface part with $\pi$ symmetry only.
