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Recent progress in deep generative models has improved the quality of voice conversion in the speech domain. However, high-quality singing voice conversion (SVC) of unseen singers remains challenging due to the wider variety of musical expressions in pitch, loudness, and pronunciation. Moreover, singing voices are often recorded with reverb and accompaniment music, which make SVC even more challenging. In this work, we present a robust one-shot SVC (ROSVC) that performs any-to-any SVC robustly even on such distorted singing voices. To this end, we first propose a one-shot SVC model based on generative adversarial networks that generalizes to unseen singers via partial domain conditioning and learns to accurately recover the target pitch via pitch distribution matching and AdaIN-skip conditioning. We then propose a two-stage training method called Robustify that train the one-shot SVC model in the first stage on clean data to ensure high-quality conversion, and introduces enhancement modules to the encoders of the model in the second stage to enhance the feature extraction from distorted singing voices. To further improve the voice quality and pitch reconstruction accuracy, we finally propose a hierarchical diffusion model for singing voice neural vocoders. Experimental results show that the proposed method outperforms state-of-the-art one-shot SVC baselines for both seen and unseen singers and significantly improves the robustness against distortions.