  Within the standard models of particle physics and cosmology we have
calculated the big-bang prediction for the primordial abundance of \he to a
theoretical uncertainty of less than $0.1 \pct$ $(\delta Y_P < \pm 0.0002)$,
improving the current theoretical precision by a factor of 10. At this accuracy
the uncertainty in the abundance is dominated by the experimental uncertainty
in the neutron mean lifetime, $\tau_n = 885.4 \pm 2.0 sec$. The following
physical effects were included in the calculation: the zero and
finite-temperature radiative, Coulomb and finite-nucleon-mass corrections to
the weak rates; order-$\alpha$ quantum-electrodynamic correction to the plasma
density, electron mass, and neutrino temperature; and incomplete neutrino
decoupling. New results for the finite-temperature radiative correction and the
QED plasma correction were used. In addition, we wrote a new and independent
nucleosynthesis code designed to control numerical errors to be less than
0.1\pct. Our predictions for the \EL[4]{He} abundance are presented in the form
of an accurate fitting formula. Summarizing our work in one number, $ Y_P(\eta
= 5\times 10^{-10}) = 0.2462 \pm 0.0004 (expt) \pm < 0.0002 (theory)$. Further,
the baryon density inferred from the Burles-Tytler determination of the
primordial D abundance, $\Omega_B h^2 = 0.019\pm 0.001$, leads to the
prediction: $Y_P = 0.2464 \pm 0.0005 (D/H) \pm < 0.0002 (theory) \pm 0.0005
(expt)$. This ``prediction'' and an accurate measurement of the primeval \he
abundance will allow an important consistency test of primordial
nucleosynthesis.
