  This paper reports the evaluation of the tenth-order QED contribution to the
lepton g-2 from the gauge-invariant set of 2072 Feynman diagrams, called Set
IV, which are obtained by inserting a second-order lepton vacuum-polarization
loop into 518 eighth-order vertex diagrams of four-photon exchange type. The
numerical evaluation is carried out by the adaptive-iterative Monte-Carlo
integration routine vegas using the fortran codes written by the automatic
code-generating algorithm gencodeN. Some of the numerical results are confirmed
by comparison with the values of corresponding integrals that have been
obtained previously by a different method. The result for the mass-independent
contribution of the Set IV to the electron g-2 is $-7.7296 (48)
(\alpha/\pi)^5$. There is also a small mass-dependent contribution to the
electron g-2 due to the muon loop: $-0.01136 (7) (\alpha/\pi)^5$. The
contribution of the tau-lepton loop is $-0.0000937(104)(\alpha/\pi)^5$. The sum
of all these contributions to the electron g-2 is $-7.7407(49) (\alpha/\pi)^5$.
The same set of diagrams enables us to evaluate the contributions to the muon
g-2 from the electron loop, muon loop, and tau-lepton loop. They add up to
$-46.95 (17) (\alpha/\pi)^5$.
