In this paper we describe the Einstein-Kahler metric for the Cartan-Hartogs of the first type which is the special case of the Hua domains. Firstly, we reduce the Monge-Ampere equation for the metric to an ordinary differential equation in the auxiliary function X = X(z, w) = $$mid$w$mid$^2[det(I-ZZ^{T}]^{frac{1}{K}}$</TEX> (see below). This differential equation can be solved to give an implicit function in Χ. Secondly, we get the estimate of the holomorphic section curvature under the complete Einstein-K$ddot{a}$hler metric on this domain.