We report the first observation of the exclusive decay process B-->J/psi K(1)(1270) using a sample of 11.2M BB macro meson pairs collected in the Belle detector at the KEKB asymmetric energy e(+)e(-) collider. We measure branching fractions of B[B(0)-->J/psi K(0)(1)(1270)] = (1.30+/-0.34+/-0.32) x 10(-3) and B[B(+)-->J/psi K(+)(1)(1270)] = (1.80+/-0.34+/-0.39) x 10(-3), where the first error is statistical and the second is systematic. These modes constitute approximately 15% of the total number of B-->J/psi X decays. No evidence is seen for B-->J/psi K(1)(1400) and we set an upper limit for this branching fraction.