Given i.i.d. observations of a random vector $X \in \mathbb{R}^p$, we study the problem of estimating both its covariance matrix $\Sigma^*$, and its inverse covariance or concentration matrix \mbox{$\Theta^* = (\Sigma^*)^{-1}$.} We estimate $\Theta^*$ by minimizing an $\ell_1$-penalized log-determinant Bregman divergence; in the multivariate Gaussian case, this approach corresponds to $\ell_1$-penalized maximum likelihood, and the structure of $\Theta^*$ is specified by the graph of an associated Gaussian Markov random field.  We analyze the performance of this estimator under high-dimensional scaling, in which the number of nodes in the graph $p$, the number of edges $s$ and the maximum node degree $d$, are allowed to grow as a function of the sample size $n$. Our first result establishes consistency of our estimate $\widehat{\Theta}$ in the elementwise maximum-norm. This in turn allows us to derive convergence rates in Frobenius and spectral norms, with improvements upon existing results for graphs with maximum node degrees $\degmax = o(\sqrt{\spindex})$.  In our second result, we show that with probability converging to one, the estimate $\widehat{\Theta}$ correctly specifies the zero pattern of the concentration matrix $\Theta^*$.  We illustrate our theoretical results via simulations for various graphs and problem parameters, showing good correspondences between the theoretical predictions and behavior in simulations.