A search of the exclusive radiative decays B-->rho(770)gamma and B-0-->omega(782)gamma is performed on a sample of about 84x10(6) B (B) over bar events collected by the BABAR detector at the SLAC PEP-II asymmetric-energy e(+)e(-) storage ring. No significant signal is seen in any of the channels. We set upper limits on the branching fractions B of B(B-0-->rho(0)gamma)<1.2x10(-6), B(B+-->rho(+)gamma)<2.1x10(-6), and B(B-0-->omegagamma)<1.0x10(-6) at 90% confidence level (C.L.). Using the assumption that Gamma(B-->rhogamma)=Gamma(B+-->rho(+)gamma)=2xGamma(B-0-->rho(0)gamma), we find the combined limit B(B-->rhogamma)<1.9x10(-6), corresponding to B(B-->rhogamma)/B(B-->K(*)gamma)<0.047 at 90% C.L.
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