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Cloud cover is crucial information for many applications such as planning land observation missions from space. It remains however a challenging variable to forecast, and Numerical Weather Prediction (NWP) models suffer from significant biases, hence justifying the use of statistical post-processing techniques. In this study, ARPEGE (Météo-France global NWP) cloud cover is post-processed using a convolutional neural network (CNN). CNN is the most popular machine learning tool to deal with images. In our case, CNN allows the integration of spatial information contained in NWP outputs. We use a gridded cloud cover product derived from satellite observations over Europe as ground truth, and predictors are spatial fields of various variables produced by ARPEGE at the corresponding lead time. We show that a simple U-Net architecture produces significant improvements over Europe. Moreover, the U-Net outclasses more traditional machine learning methods used operationally such as a random forest and a logistic quantile regression. We introduced a weighting predictor layer prior to the traditional U-Net architecture which produces a ranking of predictors by importance, facilitating the interpretation of the results. Using $N$ predictors, only $N$ additional weights are trained which does not impact the computational time, representing a huge advantage compared to traditional methods of ranking (permutation importance, sequential selection, ...).