We present a measurement of the time-dependent CP-violating asymmetries in B0-->K(*0)gamma(K(*0)-->K(0)(S)pi(0)) decays based on 124 x 10(6) Upsilon(4S)-->BB decays collected with the BABAR detector at the PEP-II asymmetric-energy B Factory at the Stanford Linear Accelerator Center. In a sample containing 105+/-14 signal decays, we measure S(K(*)gamma)=0.25+/-0.63+/-0.14 and C(K(*)gamma)=-0.57+/-0.32+/-0.09, where the first error is statistical and the second, systematic.