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The standard Capon beamformer (SCB) achieves the maximum output signal-to-interference-plus-noise ratio in the error-free case. However, estimation errors of the signal steering vector and the array covariance matrix can result in severe performance deteriorations of the SCB, especially if the training data contains the desired signal component. A popular technique to improve the robustness against model errors is to compute the Capon beamformer with the maximum output power, considering an uncertainty set for the signal steering vector. However, maximizing the total beamformer output power may result in an insufficient suppression of interferers and noise. As an alternative approach to mitigate the detrimental effect of model errors, we propose to compute the Capon beamformer with the minimum sensitivity, considering the uncertainty set for the signal steering vector. The proposed maximally robust Capon beamformer (MRCB) is at least as robust as the maximum output power Capon beamformer with the same uncertainty set for the signal steering vector. We show that the MRCB can be implemented efficiently using Lagrange duality. Simulation results demonstrate that the MRCB outperforms state-of-the-art robust adaptive beamformers in many scenarios.