  We show that if a compact connected $n$-dimensional manifold $M$ has a
conformal class containing two non-homothetic metrics $g$ and $\tilde
g=e^{2\varphi}g$ with non-generic holonomy, then after passing to a finite
covering, either $n=4$ and $(M,g,\tilde g)$ is an ambik\"ahler manifold, or
$n\ge 6$ is even and $(M,g,\tilde g)$ is obtained by the Calabi Ansatz from a
polarized Hodge manifold of dimension $n-2$, or both $g$ and $\tilde g$ have
reducible holonomy, $M$ is locally diffeomorphic to a product $M_1\times
M_2\times M_3$, the metrics $g$ and $\tilde g$ can be written as
$g=g_1+g_2+e^{-2\varphi}g_3$ and $\tilde g=e^{2\varphi}(g_1+g_2)+g_3$ for some
Riemannian metrics $g_i$ on $M_i$, and $\varphi$ is the pull-back of a
non-constant function on $M_2$.
