  The excitation curve of e+e- annihilation into four charged pions in the
rho(770) region is calculated using three existing models with rho mesons and
pions in intermediate states supplemented by Feynman diagrams with the
a_1(1260)pi intermediate states. A two-term phenomenological Lagrangian of the
a_1-rho-pi interaction is used. The mixing angle is determined by fitting the
e+e- --> pi+pi-pi+pi- cross section data of the Novosibirsk CMD-2 collaboration
and also its combination with the low-energy part of the BaBar collaboration
data. It is shown that the inclusion of the a_1-pi intermediate states succeeds
in obtaining a good agreement with the data on both cross section and the rho^0
--> pi+pi-pi+pi- decay width. When moving to energies above 1 GeV, the
rho(1450) and rho(1700) resonances are taken into account to get excellent
agreement with the BaBar data over the full energy range up to 4.5 GeV.
