  We study fully supersymmetric $AdS_6$ vacua of half-maxi\-mal $N=(1,1)$
gauged supergravity in six space-time dimensions coupled to $n$ vector
multiplets. We show that the existence of $AdS_6$ backgrounds requires that the
gauge group is of the form $G'\times G"\subset SO(4,n)$ where $G'\subset
SO(3,m)$ and $G"\subset SO(1,n-m)$. In the $AdS_6$ vacua this gauge group is
broken to its maximal compact subgroup $SO(3)\times H'\times H"$ where
$H'\subset SO(m)$ and $H"\subset SO(n-m)$. Furthermore, the $SO(3)$ factor is
the R-symmetry gauged by three of the four graviphotons. We further show that
the $AdS_6$ vacua have no moduli that preserve all supercharges. This is
precisely in agreement with the absence of supersymmetric marginal deformations
in holographically dual five-dimensional superconformal field theories.
