We extend the Standard Model gauge group by U(1)L_e-L_μ and introduce two scalars, a doublet and a singlet, that are charged under this new group and have lepton flavour violating couplings. Since in this model μ → e processes can only be mediated by μ → τ × τ → e interactions, bounds from μ → e transitions can be avoided while allowing for accessible new physics. We consider the case of a Z′ boson with a mass of M_Z′ ≃ 10 GeV and a gauge coupling _g′ ≃ 10^-4, which is in reach of Belle-II, and a long-lived Z′ boson with a mass of MeV ≲ M_Z′ ≲ m_μ - m_e which can be probed by searching for ℓ → ℓ′ + inv.. Neutrino masses and mixing angles can also be accounted for if sterile neutrinos are added to the spectrum.