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