This paper introduces the first fully unsupervised domain adaptation (UDA) framework for unsupervised anomaly detection (UAD). The performance of UAD techniques degrades significantly in the presence of a domain shift, difficult to avoid in a real-world setting. While UDA has contributed to solving this issue in binary and multi-class classification, such a strategy is ill-posed in UAD. This might be explained by the unsupervised nature of the two tasks, namely, domain adaptation and anomaly detection. Herein, we first formulate this problem that we call the two-fold unsupervised curse. Then, we propose a pioneering solution to this curse, considered intractable so far, by assuming that anomalies are rare. Specifically, we leverage clustering techniques to identify a dominant cluster in the target feature space. Posed as the normal cluster, the latter is aligned with the source normal features. Concretely, given a one-class source set and an unlabeled target set composed mostly of normal data and some anomalies, we fit the source features within a hypersphere while jointly aligning them with the features of the dominant cluster from the target set. The paper provides extensive experiments and analysis on common adaptation benchmarks for anomaly detection, demonstrating the relevance of both the newly introduced paradigm and the proposed approach. The code will be made publicly available.