We report on a measurement of the longitudinal polarization
fraction in the decay B^0 -> J/psi K^*0 using data collected with the Collider
Dtector at Fermilab.  B^0 mesons are reconstructed through the decay chain
B^0 -> J/psi K^*0, J/psi -> mu^+mu^-, K^*0 -> K^+pi^-.  A sample of 65 \pm 10
events is used to obtain the result Gamma_L/Gamma = 0.65 \pm 0.10 (stat)
\pm 0.04 (sys).  The first measurement of Gamma_L/Gamma in the decay
B^0_s -> J/psi phi is also presented.