  There is significant evidence suggesting that continuous global symmetries
are always gauged in quantum gravity. However, very weakly gauged symmetries
seem global to an effective field theory expansion in powers of Newton's
constant. We show that, at least for Einsteinian quantum gravity on AdS, such
extremely weak gaugings are indeed in the Swampland: Consistency with AdS black
hole thermodynamics requires the bulk gauge coupling $g^2$ not to vanish faster
than $\sim\exp(\ell^{d-1}/G)$, where $\ell$ is the $AdS_{d+1}$ radius and $G$
is Newton's constant as we take the $G\rightarrow0$ limit. This translates to a
constraint in the dual large $N$ CFT, namely, that the two-point function
coefficient of the current $C_J$ cannot grow faster than $\exp(N^2)$ in the
large $N$ limit. We also recover a previously known logarithmic relationship
between the cutoff of the effective field theory in AdS, Planck's mass, and the
AdS radius.
