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We study the conditions under which the cotangent bundle $T^*M$ of aRiemaannian manifold $(M,g)$, endowed with a K\"ahlerian structure $(G,J)$ ofgeneral natural lift type (see \cite{Druta1}), is Einstein. We first obtain ageneral natural K\"ahler-Einstein structure on the cotangent bundle $T^*M$. Inthis case, a certain parameter, $\lambda$ involved in the condition for$(T^*M,G,J)$ to be a K\"ahlerian manifold, is expressed as a rational functionof the other two, the value of the constant sectional curvature, $c$, of thebase manifold $(M,g)$ and the constant $\rho$ involved in the condition for thestructure of being Einstein. This expression of $\lambda$ is just that involvedin the condition for the K\"ahlerian manifold to have constant holomorphicsectional curvature (see \cite{Druta2}). In the second case, we obtain ageneral natural K\"ahler-Einstein structure only on $T_0M$, the bundle ofnonzero cotangent vectors to $M$. For this structure, $\lambda$ is expressed asanother function of the other two parameters, their derivatives, $c$ and$\rho$.

Kahler-Einstein Structures of General Natural Lifted Type on the Cotangent Bundles