This paper presents a novel discrete Mumford-Shah model for the simultaneous bias correction and image segmentation(SBCIS) for images with intensity inhomogeneity. The model is based on the assumption that an image can be approximated by a product of true intensities and a bias field. Unlike the existing methods, where the true intensities are represented as a linear combination of characteristic functions of segmentation regions, we employ L₀ gradient minimization to enforce a piecewise constant solution. We introduce a new neighbor term into the Mumford-Shah model to allow the true intensity of a pixel to be influenced by its immediate neighborhood. A two-stage segmentation method is applied to the proposed Mumford-Shah model. In the first stage, both the true intensities and bias field are obtained while the segmentation is done using the K-means clustering method in the second stage. Comparisons with the two-stage Mumford-Shah model show the advantages of our method in its ability in segmenting images with intensity inhomogeneity.