In this paper, a spatially constrained mixture model for the segmentation of MR brain images is presented. The novelty of this work is an edge-preserving smoothness prior which is imposed on the probabilities of the voxel labels. This prior incorporates a line process, which is modeled as a Bernoulli random variable, in order to preserve edges between tissues. The main difference with other, state of the art methods imposing priors, is that the constraint is imposed on the probabilities of the voxel labels and not onto the labels themselves. Inference of the proposed Bayesian model is obtained using variational methodology and the model parameters are computed in closed form. Numerical experiments are presented where the proposed model is favorably compared to state of the art brain segmentation methods as well as to a spatially varying Gaussian mixture model.