  Let $X$ be a Fano manifold of dimension at least $2$ and $D$ be a smooth
divisor in a multiple of the anticanonical class, $\frac1\alpha(-K_X)$ with
$\alpha>1$. It is well-known that K\"ahler-Einstein metrics on $X$ with conic
singularities along $D$ may exist only if the angle $2\pi\beta$ is bigger than
some positive limit value $2\pi\beta_*$. Under the hypothesis that the
automorphisms of $D$ are induced by the automorphisms of the pair $(X,D)$, we
prove that for $\beta>\beta_*$ close enough to $\beta_*$, such
K\"ahler-Einstein metrics do exist. We identify the limits at various scales
when $\beta\rightarrow\beta_*$ and, in particular, we exhibit the appearance of
the Tian-Yau metric of $X\setminus D$.
