The homogeneous and isotropic Friedmann--Lemaître--Robertson--Walker universes are generalized to dilaton-gravity theory for a mixture of non-interacting pressureless dust and incoherent radiation, matter being described by a phenomenological energy--momentum tensor. The Robertson diagram is presented and the various types of universes are discussed. In the solutions studied, the dilaton field is seen to exert a strong influence on the behaviour of the universe near the initial singularity, as expected.