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Partial voluming (PV) is arguably the last crucial unsolved problem in Bayesian segmentation of brain MRI with probabilistic atlases. PV occurs when voxels contain multiple tissue classes, giving rise to image intensities that may not be representative of any one of the underlying classes. PV is particularly problematic for segmentation when there is a large resolution gap between the atlas and the test scan, e.g., when segmenting clinical scans with thick slices, or when using a high-resolution atlas. In this work, we present PV-SynthSeg, a convolutional neural network (CNN) that tackles this problem by directly learning a mapping between (possibly multi-modal) low resolution (LR) scans and underlying high resolution (HR) segmentations. PV-SynthSeg simulates LR images from HR label maps with a generative model of PV, and can be trained to segment scans of any desired target contrast and resolution, even for previously unseen modalities where neither images nor segmentations are available at training. PV-SynthSeg does not require any preprocessing, and runs in seconds. We demonstrate the accuracy and flexibility of the method with extensive experiments on three datasets and 2,680 scans. The code is available at https://github.com/BBillot/SynthSeg.