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.
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