We address the numerical solution of infinite-dimensional inverse problems in the framework of Bayesian inference. We consider fully nonlinear, infinite-dimensional inverse problems using a Markov chain Monte Carlo (MCMC) sampling method. To address the challenges of sampling high-dimensional probability density functions (pdfs) arising upon discretization of Bayesian inverse problems governed by PDEs, we build upon the stochastic Newton MCMC method. This method exploits problem structure by taking as a proposal density a local Gaussian approximation of the posterior pdf. The construction of the covariance is made tractable by invoking a low-rank approximation of the data misfit component of the Hessian. We introduce an approximation of the stochastic Newton proposal in which we compute the low-rank-based Hessian at just the maximum a posteriori (MAP) point, and then reuse this Hessian at each MCMC step. We compare the performance of the proposed method to the original stochastic Newton MCMC method and to an independence sampler. The comparison of the three methods is conducted on a synthetic ice sheet inverse problem. For this problem, the stochastic Newton MCMC method with a MAP-based Hessian converges at least as rapidly as the original stochastic Newton MCMC method, but is far cheaper since it avoids recomputing the Hessian at each step. While it is more expensive per sample than the independence sampler, its convergence is significantly more rapid, and thus overall it is much cheaper. Finally, we present an analysis and interpretation of the posterior distribution and classify directions in parameter space based on the extent to which they are informed by the prior or by the observations.