The electron affinity (EA) calculations based on the equation-of-motion coupled cluster method proved to be an efficient scheme in the treatment of potential energy curves (PECs) for alkali molecular ions, Me-2(+). The EA approach provides description of states obtained by an attachment of one electron to the reference, which for the Me-2(+) is a doubly ionized Me-2(+2) system. The latter has a very concrete advantage in the calculations of the PECs, since it dissociates into the closed-shell fragments (Me-2(+2) -> Me+ + Me+); hence, the restricted Hartree-Fock reference can be used in the whole range of interatomic distances. In this work accurate PECs and spectroscopic constants are obtained for the six lowest-lying electronic states of the K-2(+) ion. The relativistic effects are included by adding appropriate terms of the Douglas-Kroll Hamiltonian to the one-electron integrals.