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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%. 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