  Several decades after the discovery of superconductivity in bismuthates, the
strength of their electron-phonon coupling and its evolution with doping remain
puzzling. To clarify these issues, polycrystalline hole-doped
Ba$_{1-x}$K$_{x}$BiO$_3$ ($0.1 \le x \le 0.6$) samples were systematically
synthesized and their bulk- and microscopic superconducting properties were
investigated by means of magnetic susceptibility and muon-spin
rotation/relaxation ($\mu$SR), respectively. The phase diagram of
Ba$_{1-x}$K$_{x}$BiO$_3$ was reliably extended up to $x = 0.6$, which is still
found to be a bulk superconductor. The lattice parameter $a$ increases linearly
with K-content, implying a homogeneous chemical doping. The low-temperature
superfluid density, measured via transverse-field (TF)-$\mu$SR, indicates an
isotropic fully-gapped superconducting state with zero-temperature gaps
$\Delta_0/k_\mathrm{B}T_c$ = 2.15, 2.10, and 1.75, and magnetic penetration
depths $\lambda_0$ = 219, 184, and 279 nm for $x$ = 0.3, 0.4, and 0.6,
respectively. A change in the superconducting gap, from a nearly ideal BCS
value (1.76 $k_\mathrm{B}$$T_c$ in the weak coupling case) in the overdoped $x$
= 0.6 region, to much higher values in the optimally-doped case, implies a
gradual decrease in electron-phonon coupling with doping.
