  We study a new s-process path through an isomer of $^{186}$Re to improve a
$^{187}$Re-$^{187}$Os nucleo-cosmochronometer. The nucleus $^{187}$Re is
produced by this new path of
$^{185}$Re(n,$\gamma$)$^{186}$Re$^m$(n,$\gamma$)$^{187}$Re. We measure a ratio
of neutron capture cross-sections for the $^{185}$Re(n,$\gamma$)$^{186}$Re$^m$
and $^{185}$Re(n,$\gamma$)$^{186}$Re$^{gs}$ reactions at thermal neutron energy
because the ratio with the experimental uncertainty has not been reported.
Using an activation method with reactor neutrons, we obtain the ratio of
$R_{th}$ = 0.54 $\pm$ 0.11%. From this ratio we estimate the ratio of
Maxwellian averaged cross sections in a typical s-process environment at $kT$ =
30 keV with a help of the temperature dependence given in a statistical-model
calculation because the energy dependence of the isomer/ground ratio is smaller
than the absolute neutron capture cross-section. The ratio at $kT$=30 keV is
estimated to be $R_{st}$ = 1.3 $\pm$ 0.8%. We calculate the s-process
contribution from the new path in a steady-flow model. The additional abundance
of $^{187}$Re through this path is estimated to be $N_{s}$ = 0.56 $\pm$ 0.35%
relative to the abundance of $^{186}$Os. This additional increase of $^{187}$Re
does not make any remarkable change in the $^{187}$Re-$^{187}$Os chronometer
for an age estimate of a primitive meteorite, which has recently been found to
be affected strongly by a single supernova r-process episode.
