  We search for a light Higgs boson ($A^0$) decaying into a $\tau^+\tau^-$ or
$\mu^+\mu^-$ pair in the radiative decays of $\Upsilon(1S)$. The production of
$\Upsilon(1S)$ mesons is tagged by $\Upsilon(2S)\to \pi^+ \pi^- \Upsilon(1S)$
transitions, using 158 million $\Upsilon(2S)$ events accumulated with the Belle
detector at the KEKB asymmetric energy electron-positron collider. No
significant $A^0$ signals in the mass range from the $\tau^+\tau^-$ or
$\mu^+\mu^-$ threshold to 9.2 GeV/$c^2$ are observed. We set the upper limits
at 90\% credibility level (C.L.) on the product branching fractions for
$\Upsilon(1S)\to \gamma A^0$ and $A^0\to \tau^+\tau^-$ varying from
$3.8\times10^{-6}$ to $1.5\times10^{-4}$. Our results represent an
approximately twofold improvement on the current world best upper limits for
the $\Upsilon(1S)\to \gamma A^0(\to \tau^+\tau^-)$ production. For $A^0\to
\mu^+\mu^-$, the upper limits on the product branching fractions for
$\Upsilon(1S)\to \gamma A^0$ and $A^0\to \mu^+\mu^-$ are at the same level as
the world average limits, and vary from $3.1\times10^{-7}$ to
$1.6\times10^{-5}$. The upper limits at 90\% C.L. on the Yukawa coupling
$f_{\Upsilon(1S)}$ and mixing angle ${\rm sin}\theta_{A^0}$ are also given.
