  We calculate for the first time the form factors of the semi-leptonic decays
of the $D_s$ meson to $\eta$ and $\eta^\prime$ using lattice techniques. As a
by-product of the calculation we obtain the masses and leading distribution
amplitudes of the $\eta$ and $\eta^\prime$ mesons. We use $N_f=2+1$
non-perturbatively improved clover fermions on configurations with a lattice
spacing $a\sim 0.075$ fm. We are able to obtain clear signals for relevant
matrix elements, using several noise reduction techniques, both for the
connected and disconnected contributions. This includes a new method for
reducing the variance of pseudoscalar disconnected two-point functions. At zero
momentum transfer, we obtain for the scalar form factors, $|f_0^{D_s\to
\eta}|=0.564(11)$ and $|f_0^{D_s\to \eta'}|=0.437(18)$ at $M_\pi\approx 470\,
{\rm MeV}$, as well as $|f_0^{D_s\to \eta}|=0.542(13)$ and $|f_0^{D_s\to
\eta'}|=0.404(25)$ at $M_\pi\approx 370\, {\rm MeV}$, where the errors are
statistical only.
