  We present a lattice-QCD calculation of the $B\to\pi\ell\nu$ semileptonic
form factors and a new determination of the CKM matrix element $|V_{ub}|$. We
use the MILC asqtad 2+1-flavor lattice configurations at four lattice spacings
and light-quark masses down to 1/20 of the physical strange-quark mass. We
extrapolate the lattice form factors to the continuum using staggered chiral
perturbation theory in the hard-pion and SU(2) limits. We employ a
model-independent $z$ parameterization to extrapolate our lattice form factors
from large-recoil momentum to the full kinematic range. We introduce a new
functional method to propagate information from the chiral-continuum
extrapolation to the $z$ expansion. We present our results together with a
complete systematic error budget, including a covariance matrix to enable the
combination of our form factors with other lattice-QCD and experimental
results. To obtain $|V_{ub}|$, we simultaneously fit the experimental data for
the $B\to\pi\ell\nu$ differential decay rate obtained by the BaBar and Belle
collaborations together with our lattice form-factor results. We find
$|V_{ub}|=(3.72\pm 0.16)\times 10^{-3}$ where the error is from the combined
fit to lattice plus experiments and includes all sources of uncertainty. Our
form-factor results bring the QCD error on $|V_{ub}|$ to the same level as the
experimental error. We also provide results for the $B\to\pi\ell\nu$ vector and
scalar form factors obtained from the combined lattice and experiment fit,
which are more precisely-determined than from our lattice-QCD calculation
alone. These results can be used in other phenomenological applications and to
test other approaches to QCD.

High Energy Physics - Lattice;High Energy Physics - Experiment;High Energy Physics - Phenomenology