We report the first observation of the radiative decay of the Upsilon(1S) into a charmonium state. The significance of the observed signal of Upsilon(1S) -> gamma chi(c1) is 6.3 standard deviations including systematics. The branching fraction is calculated to be B[Upsilon(1S) -> gamma chi(c1)] = [4.7(-1.8)(+2.4) (stat)(-0.5)(+0.4) (sys) x 10(-5)]. We also searched for Upsilon(1S) radiative decays into chi(c0.2) and eta(c)(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 Upsilon(2S) mass using Upsilon(1S) tagging by the Upsilon(2S) -> Upsilon(1S)pi(+)pi(-) transitions.