  In $\rm e^+ \rm e^-$ collisions, the integrated luminosity is generally
measured from the rate of low-angle Bhabha interactions $\rm e^+ e^- \to e^+
e^- $. In the published LEP results, the inferred theoretical uncertainty of
$\pm 0.061\%$ on the predicted rate is significantly larger than the reported
experimental uncertainties. We present an updated and more accurate prediction
of the Bhabha cross section in this letter, which is found to reduce the Bhabha
cross section by about $0.048\%$, and its uncertainty to $\pm 0.037\%$. When
accounted for, these changes modify the number of light neutrino species (and
its accuracy), as determined from the LEP measurement of the hadronic cross
section at the Z peak, to $N_\nu = 2.9963 \pm 0.0074$. The 20-years-old $2
\sigma$ tension with the Standard Model is gone.
