  We present strong evidence of time stretching in the peak-to-peak time scales
in the light curves of BATSE Gamma Ray Bursts (GRBs). Extensive tests are
performed on artificially dilated bursts to verify that the procedure for
extracting the peak-to-peak time scales correctly recovers the stretching of
bursts. The resulting robust algorithm is then applied to the 4B GRB database.
We derive a stretching factor of $1.92 \pm 0.13$ between the brightest burst
group ($P > 7.7 photon \cdot cm^{-2} \cdot s^{-1}$) and the dimmest burst group
($P = 1.0 \sim 1.4 photon \cdot cm^{-2} \cdot s^{-1}$) with several independent
peak-to-peak time scale definitions and they agree within uncertainties. Such
an agreement strongly supports the interpretation of the observed time
stretching as time dilation caused by the cosmological expansion, rather than
physical selection effects. We fit the result to cosmological models with
$\Lambda = 0$, $\Omega_{0}$ from 0.2 to 1.0, and contrained the standard candle
luminosity to be $L_{0}= 7.0 \pm 2.0 \pm 2.7 \times 10^{56} photons \cdot
s^{-1}$. Our luminosity value is fully consistent with the value from the
combined PVO and BATSE LogN-LogP curve with the BATSE bright bursts at low
redshifts of $z_{bright} = 0.11 \pm 0.02 \pm 0.025$. This luminosity fit is
definitely inconsistent with the the larger distance scale implied from
associating burst density with star formation rates.
