We prove that the magnetic ground state of a single monolayer Fe on W(001) is c(2x2) antiferromagnetic, i.e., a checkerboard arrangement of antiparallel magnetic moments. Real space images of this magnetic structure have been obtained with spin-polarized scanning tunneling microscopy. An out-of-plane easy magnetization axis is concluded from measurements in an external magnetic field. The magnetic ground state and anisotropy axis are explained based on first-principles calculations.