A new method two-vacua random-phase approximation (TVRPA) for the description of the two-neutrino nuclear double-beta decay is developed. The nuclear wave functions of the intermediate odd-odd nucleus are constructed as linear combinations of proton-particle neutron-hole excitations from the ground state of the initial even-even (N, Z) nucleus and, at the same time, as neutron-particle proton-hole excitations from the ground state of the final (N - 2, Z + 2) nucleus. The Ritz variational principle is used to obtain RPA-like equations. The ground state in the initial and final nucleus is approximated with HFB wave functions including proton-neutron pairing. A projection on numbers of protons and neutrons is performed additionally on the wave functions in both nuclei to restore the exact number of particles, broken by the quasi-particle transformation. The method is applied, as a preliminary test, to the 2ν double-beta decay in 4 8 2 0 Ca 2 8 . More realistic B(GT - ) and B(GT + ) Gamow-Teller strength distributions than QRPA and PQRPA are obtained. No dependence is found on the particle-particle strength parameter g p p , renormalizing Brueckner reaction matrix elements of the Bonn potential.