We consider the low-energy effective string action in four dimensions including the leading order-alpha' terms. An exact homogeneous solution is obtained. It represents a nonsingular expanding cosmological model in which the tensor fields tend to Vanish as t --> infinity. The scale factor a(t) of the very early universe in this model has the time dependence a(t)(2) = a(0)(2) + t(2). The violation of the strong energy condition of classical General Relativity to avoid the initial singularity requires that the central charge deficit of the theory be larger than a certain value. The significance of this solution is discussed.