  The specific heat of high-purity $\rm Ba_{0.68}K_{0.32}Fe_2As_2$ single
crystals with the highest reported superconducting $T_c$ = 38.5 K was studied.
The electronic specific heat, $C_p$, below $T_c$ shows two gap features, with
$\Delta_1 \approx 11$ meV and $\Delta_2 \approx 3.5$ meV obtained from an
$\alpha$-model analysis. The reduced gap value, $2\Delta^{\rm max} / k_B T_c
\approx 6.6$, the magnitude of the specific heat jump, $\Delta C_p(T_c)/T_c$,
and its slope below $T_c$ exhibit strong-coupling character. We also show that
an Eliashberg model with two hole and two electron bands gives the correct
values of $T_c$, the superconducting gaps, and the temperature dependence of
the free-energy difference.
