Summary: Recent observations have 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 the 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 the 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 increasing \(\beta\), the resulting spectrum is tilted to be flatter with more power at 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.