  The Coulomb corrections to the cross section of $e^+e^-$ pair production in
ultrarelativistic nuclear collisions are calculated in the next-to-leading
approximation with respect to the parameter $L=\ln \gamma_A\gamma_B$
($\gamma_{A,B}$ are the Lorentz factors of colliding nuclei). We found
considerable reduction of the Coulomb corrections even for large
$\gamma_A\gamma_B$ due to the suppression of the production of $e^+e^-$ pair
with the total energy of the order of a few electron masses in the rest frame
of one of the nuclei. Our result explains why the deviation from the Born
result were not observed in the experiment at SPS.
