The accelerating expansion of the Universe in the present stage is a process that will change the spectrum of relic gravitational waves. Here we present a numerical calculation for the power spectrum of relic gravitational waves in the accelerating Universe. The results show that although the overall features of the power spectrum are similar to those in the non-accelerating models, the amplitude is smaller in order of 10-1. We also find that the spectrum is very sensitive to the index β of the inflationary expansion with the scale factor a(τ) ∝ |τ|1+β. With increase of β, the resulting spectrum tends 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 parameterβ＜ -1.8. The influence of reheating followed by the inflation has been examined.