Observation of Psi(3770)-->gammachi(c1)-->gammagammaJ/Psi.

From e(+)e(-) collision data acquired with the CLEO detector at the Cornell Electron Storage Ring, we observe the non-DD(_) decay Psi(3770))-->gammachi(c1) with a statistical significance of 6.6 standard deviations, using the two-photon cascades to J/Psi and J/Psi-->l(+)l(-). We determine sigma(e(=)e(-)-->Psi(3770))xBeta(Psi(3770)-->gammachi(c1))=(18.0 +/- 3.3 +/- 2.5) pb and branching fraction Beta(Psi(3770)-->gammachi(c1)=(2.8 +/- 0.5+/-0.4) x 10(-3). We set 90% C.L. upper limits for the transition to chi(c2) (chi(c0)): sigma x Beta<5.7 pb (<282 pb) and Beta<0.9 x 10(-3) (<44 x 10(-3)). We also determine Gamma(Psi(3770)gammachi(c1))/Gamma(Psi(3770)-->pi(+)pi(-)J/Psi)=1.5 +/- 0.3 +/- 0.3 (>1.0 at 90% C.L.), which bears upon the interpretation of X(3872).