  The spin excitations from the nonmagnetic charge-ordered insulating state of
$\alpha$-(BEDT-TTF)$_2$I$_3$ at ambient pressure have been investigated by
probing the static and low-frequency dynamic spin susceptibilities via
site-selective nuclear magnetic resonance at $^{13}$C sites. The site-dependent
values of the shift and the spin-lattice relaxation rate $1/T_1$ below the
charge-ordering transition temperature ($T_{CO} \approx$ 135 K) demonstrate a
spin density imbalance in the unit cell, in accord with the charge-density
ratio reported earlier. The shift and $1/T_1$ show activated temperature
dependence with a static (shift) gap $\Delta_S \approx$ 47-52 meV and a dynamic
($1/T_1$) gap $\Delta_R \approx$ 40 meV. The sizes of the gaps are well
described in terms of a localized spin model, where spin one-half
antiferromagnetic dimer chains are weakly coupled with each other.
