  We examine a Coulomb gas consisting of $n$ identical repelling point charges
at an arbitrary inverse temperature $\beta$, subjected to a suitable external
field. We prove that the gas is effectively localized to a small neighbourhood
of the "droplet" -- the support of the equilibrium measure determined by the
external field. More precisely, we prove that the distance between the droplet
and the vacuum is with very high probability at most proportional to
$$\sqrt{\dfrac {\log n}{\beta n}}.$$ This order of magnitude is known to be
"tight" when $\beta=1$ and the external field is radially symmetric. In
addition, we prove estimates for the one-point function in a neighbourhood of
the droplet, proving in particular a fast uniform decay as one moves beyond a
distance roughly of the order $\sqrt{\frac {\log n}{\beta n}}$ from the
droplet.
