  We analyze the $^{134}_{55}$Cs$\rightarrow^{134}_{56}$Ba and
$^{135}_{55}$Cs$\rightarrow^{135}_{56}$Ba $\beta^-$ decays, which are crucial
production channels for Ba isotopes in Asymptotic Giant Branch (AGB) stars. We
reckon, from relativistic quantum mechanis, the effects of multichannel
scattering onto weak decays, including nuclear and electronic excited states
(ES) populated above $\simeq$ 10 keV, for both parent and daughter nuclei. We
find increases in the half-lives for $T>10^8$ K (by more than a factor 3 for
$^{134}$Cs) as compared to previous works based on systematics. We also discuss
our method in view of these previous calculations. An important impact on
half-lives comes from nuclear ES decays, while including electronic
temperatures yields further increases of about 20\% at energies 10-30 keV,
typical of AGB stars of moderate mass ($M \lesssim 8~M_{\odot}$). Despite
properly considering these effects, the new rates remain sensitively lower than
the TY values, implying longer half-lives at least above 8-9 keV. Our rate
predictions are in substantial accord with recent results based on the shell
model, and strongly modify branching ratios along the $s$-process path
previously adopted. With our new rate, nucleosynthesis models well account for
the isotopic admixtures of Ba in presolar SiC grains and in the Sun.
