The molecular nature of ${D}_{s0}^{*}(2317)$ and ${D}_{s1}(2460)$ have been extensively studied from the perspective of their masses, decay properties, and production rates. In this work, we study the weak decays of $B\ensuremath{\rightarrow}{\overline{D}}^{(*)}{D}_{s0}^{*}(2317)$ and $B\ensuremath{\rightarrow}{\overline{D}}^{(*)}{D}_{s1}(2460)$ by invoking triangle diagrams where the $B$ meson first decays weakly into ${\overline{D}}^{(*)}{D}_{s}^{(*)}$ and $J/\ensuremath{\psi}K({\ensuremath{\eta}}_{c}K$), and then the ${D}_{s0}^{*}(2317)$ and ${D}_{s1}(2460)$ are dynamically generated by the final-state interactions of ${D}_{s}^{(*)}\ensuremath{\eta}$ and ${D}^{(*)}K$ via exchanges of $\ensuremath{\eta}$ and ${D}^{(*)}$ mesons. The obtained absolute branching fractions of $\mathrm{Br}[B\ensuremath{\rightarrow}{\overline{D}}^{(*)}{D}_{s0}^{*}(2317)]$ are in reasonable agreement with the experimental data, while the branching fractions of $\mathrm{Br}[B\ensuremath{\rightarrow}{\overline{D}}^{(*)}{D}_{s1}(2460)]$ are smaller than the experimental central values by almost a factor of two to three. We tentatively attribute such a discrepancy to either reaction mechanisms missing in the present work or the likely existence of a relatively larger $c\overline{s}$ component in the ${D}_{s1}(2460)$ wave function.