  We measure the ratio of branching fractions for the $\Upsilon (4S)$ decays to
$B^+B^-$ and $B^0\bar{B}{}^0$ using $B^+ \rightarrow J/\psi(\ell\ell) K^+$ and
$B^0 \rightarrow J/\psi(\ell\ell) K^0$ samples, where $J/\psi(\ell\ell)$ stands
for $J/\psi \to \ell^+\ell^-$ ($\ell = e$ or $\mu$), with $711$ fb$^{-1}$ of
data collected at the $\Upsilon(4S)$ resonance with the Belle detector. We find
the decay rate ratio of $\Upsilon(4S) \rightarrow B^+B^-$ over $\Upsilon(4S)
\rightarrow B^0\bar{B}{}^0$ to be $1.065\pm0.012\pm 0.019 \pm 0.047$, which is
the most precise measurement to date. The first and second uncertainties are
statistical and systematic, respectively, and the third uncertainty is
systematic due to the assumption of isospin symmetry in $B \to J/\psi(\ell\ell)
K$.
