  This work deals with the computation of electron pair correction to small
angle Bhabha scattering, in order to contribute to the improvement of
luminometry precision at LEP/SLC below 0.1% theoretical accuracy. The exact QED
four-fermion matrix element for $e^+e^-\to e^+e^-e^+e^-$, including all
diagrams and mass terms, is computed and different Feynman graph topologies are
studied to quantify the error of approximate calculations present in the
literature. Several numerical results, obtained by a Monte Carlo program with
full matrix element, initial-state radiation via collinear structure functions,
and realistic event selections, are shown and critically compared with the
existing ones. The present calculation, together with recent progress in the
sector of $O(\alpha^2)$ purely photonic corrections, contributes to achieve a
total theoretical error in luminometry at the 0.05% level, close to the current
experimental precision and important in view of the final analysis of the
electroweak precision data.
