From e{sup +}e{sup -} collision data acquired with the CLEO detector at the Cornell Electron Storage Ring, we observe the non-DD decay {psi}(3770){yields}{gamma}{chi}{sub c1} with a statistical significance of 6.6 standard deviations, using the two-photon cascades to J/{psi} and J/{psi}{yields}l{sup +}l{sup -}. We determine {sigma}(e{sup +}e{sup -}{yields}{psi}(3770))xB({psi}(3770){yields}{gamma}{chi}{sub c1})=(18.0{+-}3.3= {+-}2.5) pb and branching fraction B({psi}(3770){yields}{gamma}{chi}{sub c1})=(2.8{+-}0.5{+-}0.4)x10{sup -3}. We set 90% C.L. upper limits for the transition to {chi}{sub c2} ({chi}{sub c0}): {sigma}xB<5.7 pb (<282 pb) and B<0.9x10{sup -3} (<44x10{sup -3}). We also determine {gamma}({psi}(3770){yields}{gamma}{chi}{sub c1})/{gamma}({psi}(3770){yields}{pi}{sup +}{pi}{sup -}J/= {psi})=1.5{+-}0.3{+-}0.3 (>1.0 at 90% C.L.), which bears upon the interpretation of X(3872)