The electronic absorption spectrum of the naphthalene cation has been obtained in conditions relevant for comparison with the diffuse interstellar bands in astrophysics, i.e., cold species in the gas phase. The novel technique consisting to photodissociate a selectively R2P2CI-prepared PAH-argon van der Waals complex in a molecular beam [Ph. Brechignac and T. Pino, Astron. Astrophys. 343, L49 (1999)] has been used. The various aspects of the method are described in detail. The whole visible range has been explored revealing two electronic transitions displaying 28 vibronic bands. Absolute absorption cross sections have also been measured, and found much larger than reported from rare gas matrices studies. The additional information on the matrix-induced or complex-induced shifts and widths, and on the intramolecular and intermolecular processes involved in these species, is discussed. No definite conclusion about the possible presence of the cation in space can be drawn so far.