  We report the first observation of the exclusive decay process B->J/psi
K1(1270) using a sample of 11.2M BBbar meson pairs collected in the Belle
detector at the KEKB asymmetric e+e- collider. We measure branching fractions
of Bf(B0 -> J/psi K10(1270) = (1.30 +-0.34 +-0.31)x10^-3 and Bf(B+ -> J/psi
K1+(1270) = (1.80 +-0.34 +-0.39)x10-3, where the first error is statistical and
the second systematic. These modes constitute approximately 15% of the total
number of B -> J/psiX decays. No evidence is seen for B -> J/psi K1(1400) and
we set an upper limit on this branching fraction. The K1(1270) -> K0pi+pi-
decays have a substantial K0rho0 intermediate state component that may be
useful for CP violation studies.
