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