  The QED contribution to the dipole polarizability of the $^4$He atom was
computed, including the effect of finite nuclear mass. The computationally most
challenging contribution of the second electric-field derivative of the Bethe
logarithm was obtained using two different methods: the integral representation
method of Schwartz and the sum-over-states approach of Goldman and Drake. The
results of both calculations are consistent, although the former method turned
out to be much more accurate. The obtained value of the electric-field
derivative of the Bethe logarithm, equal to $0.048\,557\,2(14)$ in atomic
units, confirms the small magnitude of this quantity found in the only previous
calculation [G. {\L}ach, B. Jeziorski, and K. Szalewicz, Phys. Rev. Lett. 92,
233001 (2004)], but differs from it by about 5\%. The origin of this difference
is explained. The total QED correction of the order of {\alpha} 3 in the
fine-structure constant {\alpha} amounts to 30.6671(1)$\cdot 10^{-6}$,
including the 0.1822$\cdot 10^{-6}$ contribution from the electric-field
derivative of the Bethe logarithm and the 0.01112(1)$\cdot 10^{-6}$ correction
for the finite nuclear mass, with all values in atomic units. The resulting
theoretical value of the molar polarizability of helium-4 is
$0.517\,254\,08(5)\,$cm$^3$/mol with the error estimate dominated by the
uncertainty of the QED corrections of order $\alpha^4$ and higher. Our value is
in agreement with but an order of magnitude more accurate than the result
$0.517\, 254\, 4(10)\,$cm$^3$/mol of the most recent experimental determination
[C. Gaiser and B. Fellmuth, Phys. Rev. Lett. 120, 123203 (2018)].
