This letter discusses magnon-mediated superconductivity in ferromagnetic metals. The mechanism explains in a natural way the fact that the superconductivity in UGe₂, ZrZn₂ and URhGe is apparently confined to the ferromagnetic phase. The order parameter is a spin anti-parallel component of a spin-1 triplet with zero spin projection. The transverse spin fluctuations are pair forming and the longitudinal ones are pair breaking. The competition between magnons and paramagnons explains the existence of two successive quantum phase transitions in UGe₂, from ferromagnetism to ferromagnetic superconductivity, and at higher pressure to paramagnetism. The maximum T_SC results from the suppression of the paramagnon contribution. To form a Cooper pair an electron transfers from one Fermi surface to the other. As a result, the onset of superconductivity leads to the appearance of two Fermi surfaces in each of the spin up and spin down momentum distribution functions. This fact explains the linear temperature dependence at low temperature of the specific heat, and the experimental results for UGe₂.