  In this article we study almost contact manifolds admitting weakly Einstein
metrics. We first prove that if a (2n+1)-dimensional Sasakian manifold admits a
weakly Einstein metric then its scalar curvature $s$ satisfies $-6\leqslant s
\leqslant 6$ for $n=1$ and $-2n(2n+1)\frac{4n^2-4n+3}{4n^2-4n-1}\leqslant s
\leqslant 2n(2n+1)$ for $n\geqslant2$. Secondly, for a (2n+1)-dimensional
weakly Einstein contact metric $(\kappa,\mu)$-manifold with $\kappa<1$, we
prove that it is flat or is locally isomorphic to the Lie group $SU(2)$,
$SL(2)$, or $E(1,1)$ for $n=1$ and that for $n\geqslant2$ there are no weakly
Einstein metrics on contact metric $(\kappa,\mu)$-manifolds with $0<\kappa<1$.
For $\kappa<0$, we get a classification of weakly Einstein contact metric
$(\kappa,\mu)$-manifolds. Finally, it is proved that a weakly Einstein almost
cosymplectic $(\kappa,\mu)$-manifold with $\kappa<0$ is locally isomorphic to a
solvable non-nilpotent Lie group.
