Maximum likelihood estimation furnishes powerful insights into voting theory, and the design of voting rules. However the MLE can usually be badly corrupted by a single outlying sample. This means that a single voter or a group of colluding voters can vote strategically and drastically affect the outcome. Motivated by recent progress in algorithmic robust statistics, we revisit the fundamental problem of estimating the central ranking in a Mallows model, but ask for an estimator that is provably robust, unlike the MLE. Our main result is an efficiently computable estimator that achieves nearly optimal robustness guarantees. In particular the robustness guarantees are dimension-independent in the sense that our overall accuracy does not depend on the number of alternatives being ranked. As an immediate consequence, we show that while the landmark Gibbard-Satterthwaite theorem tells us a strong impossiblity result about designing strategy-proof voting rules, there are quantitatively strong ways to protect against large coalitions if we assume that the remaining voters voters are honest and their preferences are sampled from a Mallows model. Our work also makes technical contributions to algorithmic robust statistics by designing new spectral filtering techniques that can exploit the intricate combinatorial dependencies in the Mallows model.