  The $^{22}$Ne(p,$\gamma$)$^{23}$Na reaction takes part in the neon-sodium
cycle of hydrogen burning. This cycle is active in asymptotic giant branch
stars as well as in novae and contributes to the nucleosythesis of neon and
sodium isotopes. In order to reduce the uncertainties in the predicted
nucleosynthesis yields, new experimental efforts to measure the
$^{22}$Ne(p,$\gamma$)$^{23}$Na cross section directly at the astrophysically
relevant energies are needed. In the present work, a feasibility study for a
$^{22}$Ne(p,$\gamma$)$^{23}$Na experiment at the Laboratory for Underground
Nuclear Astrophysics (LUNA) 400\,kV accelerator deep underground in the Gran
Sasso laboratory, Italy, is reported. The ion beam induced $\gamma$-ray
background has been studied. The feasibility study led to the first observation
of the $E_{\rm p}$ = 186\,keV resonance in a direct experiment. An experimental
lower limit of 0.12\,$\times$\,10$^{-6}$\,eV has been obtained for the
resonance strength. Informed by the feasibility study, a dedicated experimental
setup for the $^{22}$Ne(p,$\gamma$)$^{23}$Na experiment has been developed. The
new setup has been characterized by a study of the temperature and pressure
profiles. The beam heating effect that reduces the effective neon gas density
due to the heating by the incident proton beam has been studied using the
resonance scan technique, and the size of this effect has been determined for a
neon gas target.
