  We develop an exact functional method applied to the bosonic string on a
shperical world sheet, in graviton and dilaton backgrounds, consistent with
conformal invariance. In this method, quantum fluctuations are controled by the
amplitude of the kinetic term of the corresponding stringy sigma-model, and we
exhibit a novel non-perturbative non-critical string configuration which
appears as a fixed point of our evolution equation. We argue that this string
configuration is an exact solution, valid to all orders in alpha', which is
consistent with string scattering amplitudes. The dilaton configuration is
logarithmic in terms of the string coordinate X^0, and the amplitude of the
corresponding quantum fluctuations is independent of the target space dimension
D; for D=4, the corresponding Universe, in the Einstein frame, is static and
flat. A linearization around this fixed point leads to a slowly expanding,
decelerating Universe, reaching asymptotically (in Einstein time) the Minkowski
Universe. Moreover, the well-known linear (in terms of X^0) dilaton background,
which is a trivial fixed point of our evolution equation, is recovered by our
non trivial fixed point for early times. This feature explains the time
evolution from a linearly expanding Universe to a Minkowski Universe.
