  In recent work, we showed that non-perturbative vacuum effects of a very low
mass particle could induce, at a redshift of order 1, a transition from a
matter-dominated to an accelerating universe. In that work, we used the
simplification of a sudden transition out of the matter-dominated stage and
were able to fit the Type Ia supernovae (SNe-Ia) data points with a
spatially-open universe. In the present work, we find a more accurate, smooth
{\it spatially-flat} analytic solution to the quantum-corrected Einstein
equations. This solution gives a good fit to the SNe-Ia data with a particle
mass parameter $m_h$ in the range $6.40 \times 10^{-33}$ eV to $7.25 \times
10^{-33}$ eV. It follows that the ratio of total matter density (including dark
matter) to critical density, $\O_0$, is in the range 0.58 to 0.15, and the age
$t_0$ of the universe is in the range $8.10 h^{-1}$ Gyr to $12.2 h^{-1}$ Gyr,
where $h$ is the present value of the Hubble constant, measured as a fraction
of the value 100 km/(s Mpc). This spatially-flat model agrees with estimates of
the position of the first acoustic peak in the small angular scale fluctuations
of the cosmic background radiation, and with light-element abundances of
standard big-bang nucleosynthesis. Our model has only a single free parameter,
$m_h$, and does not require that we live at a special time in the evolution of
the universe.
