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_o~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.
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