Localization of cortical auditory evoked potentials in cochlear implant (CI) users is confounded by the presence of a stimulus artifact produced by the implant. Linearly constrained minimum variance (LCMV) beamformers are a class of adaptive spatial filters that localize sources of interest by minimizing the contributions of other uncorrelated sources. We have developed an artifact suppression method that enables an LCMV beamformer to reconstruct cortical activity with minimal artifact interference. This is accomplished by formulating the beamformer to enforce zero-gain on sources that generate the artifact lead potential. The artifact suppression method is first applied to normal hearing subject data to verify that it does not significantly distort the measured cortical responses. The effectiveness of the method is then demonstrated using simulated data and electroencephalography data from a CI user.