  We have completed the evaluation of all mass-dependent $\alpha^4$ QED
contributions to the muon $g-2$, or $a_\mu$, in two or more different
formulations. Their numerical values have been greatly improved by an extensive
computer calculation. The new value of the dominant $\alpha^4$ term $A_2^{(8)}
(m_\mu / m_e) $ is 132.6823 (72), which supersedes the old value 127.50 (41).
The new value of the three-mass term $A_3^{(8)} (m_\mu / m_e, m_\mu / m_\tau) $
is 0.0376 (1). The term $A_2^{(8)} (m_\mu / m_\tau) $ is crudely estimated to
be about 0.005 and may be ignored for now. The total QED contribution to
$a_\mu$ is $116 584 719.58 (0.02)(1.15)(0.85) \times 10^{-11}$, where 0.02 and
1.15 are uncertainties in the $\alpha^4$ and $\alpha^5$ terms and 0.85 is from
the uncertainty in $\alpha$ measured by atom interferometry. This raises the
Standard Model prediction by $13.9 \times 10^{-11}$, or about 1/5 of the
measurement uncertainty of $a_\mu$. It is within the noise of current
uncertainty ($\sim 100 \times 10^{-11}$) in the estimated hadronic
contributions to $a_\mu$.
