Conformally related Riemannian metrics with non-generic holonomy

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 $M_1\times M_2\times M_3$,themetrics$g$and$\tilde g$canbewrittenas$g=g_1+g_2+e^{-2\varphi}g_3$and$\tilde g=e^{2\varphi}(g_1+g_2)+g_3$ for some Riemannian metrics $g_i$ on $M_i$, and $\phi$ is the pull-back of a non-constant function on $M_2$.