The problem of density estimation aims to find an unknown density p on R^d given i.i.d. samples from it. In this work we study how to solve this problem assuming that p is log-concave and obeys certain dependence structure — that of an undirected graphical model. This is an instance of “nonparametric density estimation”, a challenging problem in theoretical statistics. We show that the maximum likelihood estimator (MLE) for p exists and is unique with probability 1. Furthermore, we precisely describe the support of the MLE, and we find a finite dimensional convex optimization problem that computes the MLE in the case of chordal graphs. We conclude with a discussion of the statistical complexity of this problem.