Since the discovery of superconductivity, there has been a drive to understand the mechanisms by which it occurs. The BCS (Bardeen-Cooper-Schrieffer) model successfully treats the electrons in conventional superconductors as pairs coupled by phonons (vibrational modes of oscillation) moving through the material, but there is as yet no accepted model for high-transition-temperature, organic or 'heavy fermion' superconductivity. Experiments that reveal unusual properties of those superconductors could therefore point the way to a deeper understanding of the underlying physics. In particular, the response of a material to a magnetic field can be revealing, because this usually reduces or quenches superconductivity. Here we report measurements of the heat capacity and magnetization that show that, for particular orientations of an external magnetic field, superconductivity in the heavy-fermion material CeCoIn(5) is enhanced through the magnetic moments (spins) of individual electrons. This enhancement occurs by fundamentally altering how the superconducting state forms, resulting in regions of superconductivity alternating with walls of spin-polarized unpaired electrons; this configuration lowers the free energy and allows superconductivity to remain stable. The large magnetic susceptibility of this material leads to an unusually strong coupling of the field to the electron spins, which dominates over the coupling to the electron orbits.