  We compute the zero-recoil form factor for the semileptonic decay
$\bar{B}^0\to D^{*+}\ell^-\bar{\nu}$ (and modes related by isospin and charge
conjugation) using lattice QCD with three flavors of sea quarks. We use an
improved staggered action for the light valence and sea quarks (the MILC
\asqtad\ configurations), and the Fermilab action for the heavy quarks. Our
calculations incorporate higher statistics, finer lattice spacings, and lighter
quark masses than our 2008 work. As a byproduct of tuning the new data set, we
obtain the $D_s$ and $B_s$ hyperfine splittings with few-MeV accuracy. For the
zero-recoil form factor, we obtain $\mathcal{F}(1)=0.906(4)(12)$, where the
first error is statistical and the second is the sum in quadrature of all
systematic errors. With the latest HFAG average of experimental results and a
cautious treatment of QED effects, we find $|V_{cb}| = (39.04 \pm
0.49_\text{expt} \pm 0.53_\text{QCD} \pm 0.19_\text{QED})\times10^{-3}$. The
QCD error is now commensurate with the experimental error.
