A measurement of the forward-backward asymmetries of e(+)e(-) --> b (b) over bar and e(+)e(-) --> c (c) over bar 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 --> l(-) and c --> l(+) 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 roots and the average B mixing parameter are determined simultaneously in the fit. Using all data collected by OPAL near the Z resonance, the asymmetries are measured to be: A(FB)(b (b) over bar) = (4.7 +/- 1.8 +/- 0.1)%, A(FB)(c (c) over bar) = (-6.8 +/- 2.5 +/- 0.9)% at = 89.51 GeV, A(FB)(b (b) over bar) = (9.72 +/- 0.42 +/- 0.15)%, A(FB)(c (c) over bar) = (5.68 +/- 0.54 +/- 0.39)% at = 91.25 GeV, A(FB)(b (b) over bar) = (10.3 +/- 1.5 +/- 0.2)%, A(FB)(c (c) over bar) = (14.6 +/- 2.0 +/- 0.8)% at = 92.95 GeV, For the average B mixing parameter, a value of: (chi) over bar = (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 sin(2)theta(eff)(l) of 0.23238 +/- 0.00052. (C) 2003 Published by Elsevier B.V.