  A search for $B_{(s)}^{0} \to K_{S}^{0} K^{*}(892)^{0}$ decays is performed
using $pp$ collision data, corresponding to an integrated luminosity of
$1.0~\text{fb}^{-1}$, collected with the LHCb detector at a centre-of-mass
energy of $7~\text{TeV}$. The $B_{s}^{0} \to K_{S}^{0} K^{*}(892)^{0}$ decay is
observed for the first time, with a significance of 7.1 standard deviations.
The branching fraction is measured to be \begin{equation*}
\mathcal{B}(B_{s}^{0} \to \bar{K}^{0} K^{*}(892)^{0}) + \mathcal{B}(B_{s}^{0}
\to K^{0} \bar{K}^{*}(892)^{0}) = (16.4 \pm 3.4 \pm 2.3) \times 10^{-6},\\
\end{equation*} where the first uncertainty is statistical and the second is
systematic. No evidence is found for the decay $B^{0} \to K_{S}^{0}
K^{*}(892)^{0}$ and an upper limit is set on the branching fraction,
$\mathcal{B}(B^{0} \to \bar{K}^{0} K^{*}(892)^{0}) + \mathcal{B}(B^{0} \to
K^{0} \bar{K}^{*}(892)^{0}) < 0.96 \ \times 10^{-6}, $ at $90\,\% $ confidence
level. All results are consistent with Standard Model predictions.
