  We report on a novel phase transition at $T$ = 0.9 K in the Ce-based
filled-skutterudite compound CeOs$_{4}$Sb$_{12}$ via measurements of the
nuclear-spin lattice relaxation rate $1/T_{1}$ and nuclear quadrupole resonance
(NQR) spectrum of Sb nuclei. The temperature ($T$) dependence of $1/T_1$
behaves as if approaching closely an antiferromagnetic (AFM) quantum critical
point (QCP), following the relation $1/T_{1}\propto T/(T-T_{\rm N})^{1/2}$ with
$T_{\rm N} = 0.06$ K in the range of $T=1.3-25$ K. The onset of either the
spin-density-wave (SDW) or charge-density-wave (CDW) order at $T_{\rm 0} = 0.9$
K, that is, of the first order, is evidenced by a broadening of the NQR
spectrum and a marked reduction in $1/T_1$ just below $T_{\rm 0}$. The
$f$-electron-derived correlated band realized in CeOs$_{4}$Sb$_{12}$ is
demonstrated to give rise to the novel phase transition on the verge of AFM
QCP.
