  We consider 5-dimensional cosmological solutions of a single brane. The
correct cosmology on the brane, i.e., governed by the standard 4-dimensional
Friedmann equation, and stable compactification of the extra dimension is
guaranteed by the existence of a non-vanishing \hat{T}^5_5 which is
proportional to the 4-dimensional trace of the energy-momentum tensor. We show
that this component of the energy-momentum tensor arises from the backreaction
of the dilaton coupling to the brane. The same positive features are exhibited
in solutions found in the presence of non-vanishing cosmological constants both
on the brane (\Lambda_{br}) and in the bulk (\Lambda_B). Moreover, the
restoration of the Friedmann equation, with the correct sign, takes place for
both signs of $\Lambda_B$ so long as the sign of $\Lambda_{br}$ is opposite
$\Lambda_B$ in order to cancel the energy densities of the two cosmological
constants. We further extend our single-brane thin-wall solution to allow a
brane with finite thickness.
