[For part I, cf. Proc. 21st Int. Taniguchi Symp., Katata/Japan, Conf., Kyoto/Japan 1987, Lect. Notes Math. 1339, 20-31 (1988; Zbl 0659.53048).] Let X be a compact Fano manifold and D be a smooth reduced divisor in X such that \(c_ 1(X)=\alpha [D]\) with \(\alpha >1\). If D admits an Einstein-Kähler metric, then there exists a complete Ricci-flat Kähler metric on \(X\setminus D\). This is a special case of the result announced by S.-T. Yau. The outline of the proof is like this: First using Calabi's construction we make a model metric which is asymptotically flat in a suitable sense. Next we employ the continuity method to deform the metric to make it Ricci-flat. Here we effectively use the Sobolev inequality and the interior Schauder estimates.