The aim of the paper is to describe the Einstein-Kähler metric for the domain \(D = \{(z,w)\in {\mathbb{C}}^ n\times {\mathbb{C}}: \| z\|^ 2 + | w|^{2p}< 1\}\) where p is a positive number. A formula for the metric is given in terms of a function determined by a differential equation. Moreover, the components of the curvature tensor for the metric are calculated and the behaviour of the metric near boundary points of special interest is analyzed. (The existence of Einstein-Kähler metrics in bounded pseudoconvex domains has been proved by N. Mok and S. T. Yau in Proc. Symp. Pure Math. 39, Part 1, 41-59 (1983; Zbl 0526.53056). The case of bounded pseudoconvex domains with \({\mathcal C}^ 2\)-boundary has been investigated by S. Y. Cheng and S. T. Yau in Commun. Pure Appl. Math. 33, 507-544 (1980; Zbl 0506.53031).)