The two-Coulomb-centre problem for the continuous spectrum is treated in prolate spheroidal coordinates. Solutions of the one-dimensional equations that are obtained after separation of spatial variables in the Schrodinger equation are found for large distances R between the Coulomb centres. The solutions are obtained in a closed algebraic form convenient for their further application. The obtained solutions present the expansions of the exact eigenvalues and eigenfunctions of the quasiradial and quasiangular equations in inverse powers of R. The derived wavefunctions allow us to investigate completely the cosmological recombination problem, namely, to include in the calculation a quasimolecular mechanism of formation of atomic hydrogen in the early universe.