  The $p_{\perp}$-distribution for a dilepton production at the LHC with high
luminosity is calculated in the Randall-Sundrum scenario with a small curvature
$\kappa$. The widths of massive gravitons are taken into account. The discovery
limits on 5-dimensional gravity scale $M_5$ are obtained to be 17.8 TeV and
21.6 TeV for $\sqrt{s} = 14$ TeV and integrated luminosities 1000 fb$^{-1}$ and
3000 fb$^{-1}$, respectively. Contrary to the standard RS model, these limits
do not depend on parameter $\kappa$.
