  The first observation of the $B_s^0 \to \overline{D}^0 K^+ K^-$ decay is
reported, together with the most precise branching fraction measurement of the
mode $B^0 \to \overline{D}^0 K^+ K^-$. The results are obtained from an
analysis of $pp$ collision data corresponding to an integrated luminosity of
$3.0~\textrm{fb}^{-1}$. The data were collected with the LHCb detector at
centre-of-mass energies of $7$ and $8$ TeV. The branching fraction of the $B^0
\to \overline{D}^0 K^+ K^-$ decay is measured relative to that of the decay
$B^0 \to \overline{D}^0 \pi^+ \pi^-$ to be $$\frac{\mathcal{B}(B^0 \to
\overline{D}^0 K^+ K^-)}{\mathcal{B}(B^0 \to \overline{D}^0 \pi^+ \pi^-)} =
(6.9 \pm 0.4 \pm 0.3)\%,$$ where the first uncertainty is statistical and the
second is systematic. The measured branching fraction of the $B_s^0 \to
\overline{D}^0 K^+ K^-$ decay mode relative to that of the corresponding $B^0$
decay is $$\frac{\mathcal{B}(B_s^0 \to \overline{D}^0 K^+ K^-)}{\mathcal{B}(B^0
\to \overline{D}^0 K^+ K^-)} = (93.0 \pm 8.9 \pm 6.9)\%.$$ Using the known
branching fraction of ${B^0 \to \overline{D}^0 \pi^+ \pi^-}$, the values of
${{\mathcal B}(B^0 \to \overline{D}^0 K^+ K^- )=(6.1 \pm 0.4 \pm 0.3 \pm 0.3)
\times 10^{-5}}$, and ${{\cal B}(B_s^0 \to \overline{D}^0 K^+ K^-)=}$ $(5.7 \pm
0.5 \pm 0.4 \pm 0.5) \times 10^{-5}$ are obtained, where the third
uncertainties arise from the branching fraction of the decay modes ${B^0 \to
\overline{D}^0 \pi^+ \pi^-}$ and $B^0 \to \overline{D}^0 K^+ K^-$,
respectively.
