The possibility that the expansion rate of the Universe, as reflected by the Red Shift, could be produced by the existence of the dilaton field is explored. The analysis starts from previously studied solutions of the Einstein equations for gravity interacting with a massive scalar field. It is firstly underlined that such solutions can produce the observed values of the Hubble constant. Since the Einstein-Klein-Gordon lagrangian could be expected to appear as an effective one for the dilaton in some approximation, the mentioned solutions are applied to study this field. Therefore, the vacuum expectation value for the dilaton is selected to be of the order of the Planck mass, as it is frequently fixed in string phenomenology. Then, it follows that the value of its effective mass should be as low as m=3.9 10^(-29) cm^(-1) in order produce the observed expansion rate. The discussion can also predict a radius of the Universe of the order of 10^(29) cm. Finally, after adopting the view advanced ina previous work, in which these mentioned solutions are associated to interior configurations of collapsed scalar fields, a picture of our Universe as a black hole interior is suggested.