This paper introduces a regularization method to ex-plicitly control the rank of a learned symmetric positive semidefinite distance matrix in distance metric learning. To this end, we propose to incorporate in the objective function a linear regularization term that minimizes the k smallest eigenvalues of the distance matrix. It is equivalent to min-imizing the trace of the product of the distance matrix with a matrix in the convex hull of rank-k projection matrices, called a Fantope. Based on this new regularization method, we derive an optimization scheme to efficiently learn the distance matrix. We demonstrate the effectiveness of the method on synthetic and challenging real datasets of face verification and image classification with relative attributes, on which our method outperforms state-of-the-art metric learning algorithms.