  Observations have recently indicated that the Universe at the present stage
is in an accelerating expansion, a process that has great implications. We
evaluate the spectrum of relic gravitational waves in the current accelerating
Universe and find that there are new features appearing in the resulting
spectrum as compared to the decelerating models. In the low frequency range the
peak of spectrum is now located at a frequency $\nu_E \simeq
(\frac{\Omega_m}{\Omega_{\Lambda}})^{1/3} \nu_H$, where $\nu_H$ is the Hubble
frequency, and there appears a new segment of spectrum between $\nu_E$ and
$\nu_H$. In all other intervals of frequencies $\geq \nu_H$, the spectral
amplitude acquires an extra factor $\frac{\Omega_m}{\Omega_{\Lambda}}$, due to
the current acceleration, otherwise the shape of spectrum is similar to that in
the decelerating models. The recent WMAP result of CMB anisotropies is used to
normalize the amplitude for gravitational waves. The slope of the power
spectrum depends sensitively on the scale factor $a(\tau) \propto
|\tau|^{1+\beta}$ during the inflationary stage with $\beta=-2$ for the exact
de Sitter space. With the increasing of $\beta$, the resulting spectrum is
tilted to be flatter with more power on high frequencies, and the sensitivity
of the second science run of the LIGO detectors puts a restriction on the
parameter $\beta \leq -1.8$. We also give a numerical solution which confirms
these features.
