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Multi-modal medical image completion has been extensively applied to alleviate the missing modality issue in a wealth of multi-modal diagnostic tasks. However, for most existing synthesis methods, their inferences of missing modalities can collapse into a deterministic mapping from the available ones, ignoring the uncertainties inherent in the cross-modal relationships. Here, we propose the Unified Multi-Modal Conditional Score-based Generative Model (UMM-CSGM) to take advantage of Score-based Generative Model (SGM) in modeling and stochastically sampling a target probability distribution, and further extend SGM to cross-modal conditional synthesis for various missing-modality configurations in a unified framework. Specifically, UMM-CSGM employs a novel multi-in multi-out Conditional Score Network (mm-CSN) to learn a comprehensive set of cross-modal conditional distributions via conditional diffusion and reverse generation in the complete modality space. In this way, the generation process can be accurately conditioned by all available information, and can fit all possible configurations of missing modalities in a single network. Experiments on BraTS19 dataset show that the UMM-CSGM can more reliably synthesize the heterogeneous enhancement and irregular area in tumor-induced lesions for any missing modalities.