We propose a novel convolutional neural network for lesion detection from weak labels. Only a single, global label per image - the lesion count - is needed for training. We train a regression network with a fully convolutional architecture combined with a global pooling layer to aggregate the 3D output into a scalar indicating the lesion count. When testing on unseen images, we first run the network to estimate the number of lesions. Then we remove the global pooling layer to compute localization maps of the size of the input image. We evaluate the proposed network on the detection of enlarged perivascular spaces in the basal ganglia in MRI. Our method achieves a sensitivity of with on average 1.5 false positives per image. Compared with four other approaches based on intensity thresholding, saliency and class maps, our method has a higher sensitivity.