  We perform a new calculation of the hadronic contributions, $a({\rm
Hadronic})$ to the anomalous magnetic moment of the muon, $a_\mu$. For the low
energy contributions of order $\alpha^2$ we carry over an analysis of the pion
form factor $F_\pi(t)$ using recent data both on $e^+e^-\to\pi^+\pi^-$ and
$\tau^+\to \bar{\nu}_\tau \pi^+\pi^0$. In this analysis we take into account
that the phase of the form factor is equal to that of $\pi\pi$ scattering. This
allows us to profit fully from analyticity properties so we can use also
experimental information on $F_\pi(t)$ at spacelike $t$. At higher energy we
use QCD to supplement experimental data, including the recent measurements of
$e^+e^-\to {\rm hadrons}$ both around 1 GeV and near the $\bar{c}c$ threshold.
This yields a precise determination of the $O(\alpha^2)$ and
$O(\alpha^2)+O(\alpha^3)$ hadronic part of the photon vacuum polarization
pieces, $$10^{11}\times a^{(2)}({\rm h.v.p.})=6 909\pm64;\quad 10^{11}\times
a^{(2+3)}({\rm h.v.p.})=7 002\pm66$$ As byproducts we also get the masses and
widths of the $\rho^0, \rho^+$, and very accurate values for the charge radius
and second coefficient of the pion. Adding the remaining order $\alpha^3$
hadronic contributions we find $$10^{11}\times a^{\rm theory}(\hbox{Hadronic})=
6 993\pm69\quad(e^+e^- + \tau + {\rm spacel.})$$ The figures given are obtained
including $\tau$ decay data. This is to be compared with the recent
experimental value, $$10^{11}\times a^{\rm exp.}(\hbox{Hadronic})=7
174\pm150.$$
