The authors pursue the classification of all normal CM-fields of class number one, a family proved to be finite by A. M. Odlyzko [Invent. Math. 29, 275–286 (1975; Zbl 0299.12010)]. The fields under examination are non-abelian of degree \(4p^2\), \(p\) an odd prime, and all the candidates in this list are found. The proof uses a combination of algebraic arguments, numerical computations and the results of S. Louboutin, R. Okazaki and M. Olivier [Trans. Am. Math. Soc. 349, 3657–3678 (1997; Zbl 0893.11045)] and of K.-Y. Chang and S.-H. Kwon [Proc. Am. Math. Soc. 128, 2517–2528 (2000; Zbl 0983.11064)]. As usual in these questions, the key point consists in reducing the field to be explored numerically by sufficiently strong theoretical arguments.