  We study the scattering of light pseudoscalar mesons ($\pi$, $K$) off charmed
mesons ($D$, $D_s$) in full lattice QCD. The S-wave scattering lengths are
calculated using L\"uscher's finite volume technique. We use a relativistic
formulation for the charm quark. For the light quark, we use domain-wall
fermions in the valence sector and improved Kogut-Susskind sea quarks. We
calculate the scattering lengths of isospin-3/2 $D\pi$, $D_s\pi$, $D_sK$,
isospin-0 $D\bar{K}$ and isospin-1 $D\bar{K}$ channels on the lattice. For the
chiral extrapolation, we use a chiral unitary approach to next-to-leading
order, which at the same time allows us to give predictions for other channels.
It turns out that our results support the interpretation of the
$D_{s0}^*(2317)$ as a $DK$ molecule. At the same time, we also update a
prediction for the isospin breaking hadronic decay width
$\Gamma(D_{s0}^*(2317)\to D_s\pi)$ to $(133\pm22)$ keV.
