  We develop a surface for the electric dipole moment of three interacting
helium atoms and use it, together with state-of-the-art potential and
polarizability surfaces, to compute the third dielectric virial coefficient,
$C_\varepsilon$, for both $^4$He and $^3$He isotopes. Our results agree with
previously published data computed using an approximated form for the
three-body polarizability, and are extended to the low-temperature regime by
including exchange effects. Additionally, the uncertainty of $C_\varepsilon$ is
rigorously determined for the first time by propagating the uncertainties of
the potential and polarizability surfaces; this uncertainty is much larger than
the contribution from the dipole-moment surface to $C_\varepsilon$. Our results
compare reasonably well with the limited experimental data. The
first-principles values of $C_\epsilon$ computed in this work will enhance the
accuracy of primary temperature and pressure metrology based on measurements of
the dielectric constant of helium.
