Early vessel wall thickening is seen as an indicator for the development of cerebrovascular disease. Quantification of wall thickening using conventional measurement methods is difficult owing to the relatively thin vessel wall thickness compared to the acquired MR voxel size. We hypothesize that a convolutional neural network (CNN), can incorporate spatial orientation, shape, and intensity distribution of the vessel wall in an accurate thickness estimation for subvoxel walls. MR imaging of 34 post-mortem specimens was performed using a 3D gradient echo protocol (isotropic acquired voxel size: 0.11 mm; acquisition time: 5h46m). Simulating clinically feasible resolutions, image patches were sampled at a clinically feasible isotropic voxel size of 0.8 mm (patch size: 113 voxels). Image patches were sampled centered around vessel wall voxels where the wall thickness of the center voxel was measured at the original resolution using a validated measurement method. The image patches were fed into our CNN, which consisted of five subsequent 3D convolutional layers, followed by two fully connected layers feeding into the linearly activated output layer. Our network can distinguish walls with a target thickness between 0.2-1.0 mm. In this range, the median offset between the target thickness and estimated thickness is 0.14 mm (interquartile range: 0.22 mm). For walls with a target thickness below and above half the voxel size (0.4 mm), the median offset is 0.17 mm and 0.10 mm, respectively. In conclusion, our results show that a CNN can accurately measure the thickness of subvoxel vessel walls, down to half the voxel size.