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) -> pi(+)pi(-)Upsilon(1S) transitions, using 158 x 10(6) 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) -> gamma A(0) and A(0) -> tau(+)tau(-) varying from 3.8 x 10(-6) to 1.5 x 10(-4). Our results represent an approximately twofold improvement on the current world best upper limits for the Upsilon(1S) -> gamma A(0) (tau(+)tau(-)) production. For A(0) -> mu(+) mu(-), the upper limits on the product branching fractions for Upsilon(1S) -> gamma A(0) and A(0) -> mu(+)mu(-) are at the same level as the world average limits, and vary from 3.1 x 10(-7) to 1.6 x 10(-5). The upper limits at 90% credibility level on the Yukawa coupling f((SIC)(1S)) and mixing angle sin theta(A0) are also given.