  The three-loop QED mass-dependent contributions to the $g-2$ of each of the
charged leptons with two internal closed fermion loops, sometimes called
$A^{(6)}_3\left(\frac{m_1}{m_2}, \frac{m_1}{m_3}\right)$ in the $g-2$
literature, is revisited using the Mellin-Barnes (MB) representation technique.
Results for the muon and $\tau$ lepton anomalous magnetic moments
$A^{(6)}_{3,\mu}$ and $A^{(6)}_{3,\tau}$, which were known as series expansions
in the lepton mass ratios up to the first few terms only, are extended to their
exact expressions. The contribution to the anomalous magnetic moment of the
electron $A^{(6)}_{3,e}$ is also explicitly given in closed form. In addition
to this, we show that the different series representations derived from the MB
representation collectively converge for all possible values of the masses.
Such unexpected behavior is related to the fact that these series bring into
play double hypergeometric series that belong to a class of Kamp\'e de F\'eriet
series which we prove to have the same simple convergence and analytic
continuation properties as the Appell $F_1$ double hypergeometric series.
