  We have calculated the stellar $\beta$-decay rate of the important s-process
branching point ${}^{134}$Cs based on the state of the art shell model
calculations. At typical $s$-process temperatures ($T\sim$ 0.2-0.3 GK), our new
rate is one order of magnitude lower than the widely-used rate from Takahashi
and Yokoi (hereafter TY87). The impact on the nucleosynthesis in AGB stars is
investigated with various masses and metallicities. Our new decay rate leads to
an overall decrease in the ${}^{134}$Ba/${}^{136}$Ba ratio, and well explains
the measured ratio in meteorities without introducing the $i$ process. We also
derive the elapsed time from the last AGB nucleosynthetic event that polluted
the early Solar System to be $>$28 Myr based on the ${}^{135}$Cs/${}^{133}$Cs
ratio, which is consistent with the elapsed times derived from ${}^{107}$Pd and
${}^{182}$Hf. The $s$-process abundance sum of ${}^{135}$Ba and ${}^{135}$Cs is
found to increase, resulting in a smaller $r$-process contribution of
${}^{135}$Ba in the Solar System.
