  Let $k$ be an imaginary quadratic field and $p$ an odd prime number such that
the $p$-rank of the class group of $k$ is one. Let $S$ be a finite set of
places of $k$ distinct from $p$-adic places. We give sufficient conditions for
the Galois group $G_S$, of the maximal pro-$p$-extension of $k$ which is
unramified outside $S$, to be \textit{mild}, hence of cohomological dimension
$2$.
