  We performed systematic thermal conductivity measurements on heavily
hole-doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$ single crystals with 0.747 $\leq x \leq$
0.974. At $x$ = 0.747, the $\kappa_0/T$ is negligible, indicating nodeless
superconducting gap. A small residual linear term $\kappa_0/T$ ($\approx$ 0.035
mW/K$^2$ cm) appears at $x$ = 0.826, and it increases slowly up to $x$ = 0.974,
followed by a drastic increase of more than 20 times to the pure KFe$_2$As$_2$
($x$ = 1.0). This doping dependence of $\kappa_0/T$ clearly shows that the
nodal gap appears near $x = 0.8$, likely associated with the change of Fermi
surface topology. The small values of $\kappa_0/T$ from $x$ = 0.826 to 0.974
support a "$\curlyvee$"-shaped nodal $s$-wave gap recently revealed by
angle-resolved photoemission spectroscopy experiments at $x$ = 0.9.
Furthermore, the drastic increase of $\kappa_0/T$ from $x$ = 0.974 to 1.0 is
inconsistent with a symmetry-imposed $d$-wave gap in KFe$_2$As$_2$, and the
possible nodal gap structure in KFe$_2$As$_2$ is discussed.
