  In this paper, we consider a quantum model of gravitation interacting with a
Born-Infeld(B-I) type scalar field $\phi$. The corresponding Wheeler-Dewitt
equation can be solved analytically for both very large and small cosmological
scale factor. In the condition that small cosmological scale factor tend to
limit, the wave function of the universe can be obtained by applying the
methods developed by Vilenkin, Hartle and Hawking. Both Vilenkin's and
Hartle-Hawking's wave function predicts nonzero cosmological constant. The
Vilenkin's wave function predicts a universe with a cosmological constant as
large as possible, while the Hartle-Hawking's wave function predicts a universe
with positive cosmological constant, which equals to $\frac{1}{\lambda}$. It is
different from Coleman's result that cosmological constant is zero, and also
different from Hawking's prediction of zero cosmological constant in quantum
cosmology with linear scalar field. We suggest that dark energy in the universe
might result from the B-I type scalar field with potential and the universe can
undergo a phase of accelerating expansion. The equation of state parameter lies
in the range of $-1<$w$<-{1/3}$. When the potential
$V(\phi)=\frac{1}{\lambda}$, our Lagrangian describes the Chaplygin gas. In
order to give a explanation to the observational results of state parameter
w$<-1$, we also investigate the phantom model that posses negative kinetic
energy. We find that weak and strong energy conditions are violated for phantom
B-I type scalar field. At last, we study a specific potential with the form
$V_0(1+\frac{\phi}{\phi_0})e^{-(\frac{\phi}{\phi_0})}$ in phantom B-I scalar
field in detail. The attractor property of the system is shown by numerical
analysis.
