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We report the first observation of the radiative decay of the Υ ( 1S ) into a charmonium state. The significance of the observed signal of Υ ( 1S ) → γχc1 is 6.3 standard deviations including systematics. The branching fraction is calculated to be B [ Υ ( 1S ) → γχc1 ] = [ 4.7-1.8+2.4 ( stat ) -0.5+0.4 ( sys ) ×10-5 ] . We also searched for Υ ( 1S ) radiative decays into χc0,2 and ηc ( 1S,2S ) , and set upper limits on their branching fractions. These results are obtained from a 24.9fb-1 data sample collected with the Belle detector at the KEKB asymmetric-energy e+e- collider at a center-of-mass energy equal to the Υ ( 2S ) mass using Υ ( 1S ) tagging by the Υ ( 2S ) → Υ ( 1S ) π+π- transitions.