  The first observation of the $D^0 \to \pi^+\pi^-\mu^+\mu^-$ and $D^0 \to
K^+K^-\mu^+\mu^-$ decays is reported using a sample of proton-proton collisions
collected by LHCb at a center-of-mass energy of 8$\,$TeV, and corresponding to
2$\,$fb$^{-1}$ of integrated luminosity. The corresponding branching fractions
are measured using as normalization the decay $D^0 \to K^-
\pi^+[\mu^+\mu^-]_{\rho^0/\omega}$, where the two muons are consistent with
coming from the decay of a $\rho^0$ or $\omega$ meson. The results are
$\mathcal{B}(D^0 \to
\pi^+\pi^-\mu^+\mu^-)=(9.64\pm0.48\pm0.51\pm0.97)\times10^{-7}$ and
$\mathcal{B}(D^0 \to K^+K^-\mu^+\mu^-)=(
1.54\pm0.27\pm0.09\pm0.16)\times10^{-7}$, where the uncertainties are
statistical, systematic, and due to the limited knowledge of the normalization
branching fraction. The dependence of the branching fraction on the dimuon mass
is also investigated.
