A measurement of the forward-backward asymmetries of e^+e^- --> Z0 --> bbbar and e^+e^- --> Z0 --> ccbar events using electrons and muons produced in semileptonic decays of bottom and charm hadrons is presented. The outputs of two neural networks designed to identify b to lepton and and c to lepton decays are used in a maximum likelihood fit to a sample of events containing one or two identified leptons. The b and c quark forward-backward asymmetries at three centre-of-mass energies sqrt(s), and the average B mixing parameter chi are determined simultaneously in the fit. Using all data collected by OPAL near the Z0 resonance, the asymmetries are measured to be
Afbb = (4.7 +- 1.8 +- 0.1)%, Afbc = (-6.8 +- 2.5 +- 0.9)% at sqrt(s) 89.51 GeV,
Afbb = (9.72 +- 0.42 +- 0.15)%, Afbc = (5.68 +- 0.54 +- 0.39)% at sqrt(s) 91.25 GeV,
Afbb = (10.3 +- 1.5 +- 0.2) %, Afbc = (14.6 +- 2.0 +- 0.8)% at sqrt(s) 92.95 GeV.
For the average B mixing parameter, a value of chi = (13.12 +- 0.49 +- 0.42)% is obtained. In each case the first uncertainty is statistical and the second systematic. These results are combined with other OPAL measurements of the b and c forward-backward asymmetries, and used to derive a value for the effective electroweak mixing angle for leptons sin2thetaeff = 0.23238 +- 0.00052.