Heavy-fermion metals exhibit a plethora of low-temperature ordering phenomena(1-4). Among these are the so-called hidden-order phases(4-8) that, in contrast-to conventional magnetic order, are invisible to standard neutron diffraction experiments. One of the structurally most simple hidden-order compounds, CeB6, has been intensively studied for an elusive phase that was attributed to the antiferroquadrupolar ordering of cerium-4f moments(9-12). As the ground state of CeB6 is characterized by a more conventional antiferromagnetic (AFM) order(9), the low-temperature physics of this system has generally been assumed to be governed solely by AFM interactions between the dipolar and multipolar Ce moments(13,14). Here we overturn this established picture by observing an intense ferromagnetic (FM) low-energy collective mode that dominates the magnetic excitation spectrum of CeB6. Inelastic neutron-scattering data reveal that the intensity of this FM excitation significantly exceeds that of conventional spin-wave magnons emanating from the AFM wavevectors, thus placing CeB6 much closer to a FM instability than previously anticipated. This propensity for ferromagnetism may account for much of the unexplained behaviour of CeB6, and should lead to a re-examination of existing theories that have so far largely neglected the role of FM interactions.