Heavy-fermion metals exhibit a plethora of low-temperature ordering phenomena, among them the so-called hidden-order phases that in contrast to conventional magnetic order are invisible to standard neutron diffraction. One of the oldest and structurally simplest hidden-order compounds, CeB6, became famous for an elusive phase that was attributed to the antiferroquadrupolar ordering of cerium-4f moments. In its ground state, CeB6 also develops a more usual antiferromagnetic (AFM) order. Hence, its essential low-temperature physics was always considered to be solely governed by AFM interactions between the dipolar and multipolar Ce moments. Here we overturn this established perspective by uncovering an intense ferromagnetic (FM) low-energy collective mode that dominates the magnetic excitation spectrum of CeB6. Our inelastic neutron-scattering data reveal that the intensity of this FM excitation by far exceeds that of conventional spin-wave magnons emanating from the AFM wave vectors, thus placing CeB6 much closer to a FM instability than could be anticipated. This propensity of CeB6 to ferromagnetism may account for much of its unexplained behavior, such as the existence of a pronounced electron spin resonance, and should lead to a substantial revision of existing theories that have so far largely neglected the role of FM interactions.