The classical evolution of a homogeneous cosmological model is investi
gated within the framework of low-energy effective string gravity with
higher-genus corrections. Various conformal frames are considered. Fo
r the general case of correction functions in the Lagrangian we derive
the cosmological solutions with arbitrary curvature and dilaton, modu
lus and Kalb-Ramond fields. These solutions are given in closed form i
n terms of integrations depending on dilaton coupling functions and ge
neralize previously known tree-level solutions. The features of the so
lutions are discussed.