Screening the electron spin of a magnetic atom via spin coupling to conduction electrons results in a strong resonant peak in the density of states at the Fermi energy, the Kondo resonance. We show that magnetic coupling of a Kondo atom to another unscreened magnetic atom can split the Kondo resonance into two peaks. Inelastic spin excitation spectroscopy with scanning tunneling microscopy is used to probe the Kondo effect of a Co atom, supported on a thin insulating layer on a Cu substrate, that is weakly coupled to a nearby Fe atom to form an inhomogeneous dimer. The Kondo peak is split by interaction with the non-Kondo atom, but can be reconstituted with a magnetic field of suitable magnitude and direction. Quantitative modeling shows that this magnetic field results in a spin-level degeneracy in the dimer, which enables the Kondo effect to occur.