A major obstacle in obtaining adjusted cross sections from integral experiments is the expensive and time-consuming evaluation of sensitivities and modeling corrections. The principal contribution of this paper is the development of a state-of-the-art Monte Carlo method that evaluates sensitivities particularly efficiently and that uses “point” nuclear data and three-dimensional combinatorial geometry to eliminate modeling errors. This method enables adjustment procedures to be applied more reliably and generally than previously possible. Theoretical advances include the way the sensitivity estimator is chosen and evaluated. Also the adjustment procedure takes into account all the Monte Carlo statistical errors, and iteration is used to cope with nonlinearities. The methods developed are successfully applied to an analysis of the Winfrith Iron Benchmark Experiment.