The b quark forward-backward asymmetry has been measured using hadronic Z0 decays collected by the OPAL experiment at LEP. Z0 -> bbbar decays were selected using a combination of secondary vertex and lepton tags, and the sign of the b quark charge was determined using an inclusive tag based on jet, vertex and kaon charges. The results, corrected to the quark level, are: A^b_FB = 0.0582 +- 0.0153 +- 0.0012 at sqrt(s) = 89.50 GeV A^b_FB = 0.0977 +- 0.0036 +- 0.0018 at sqrt(s) = 91.26 GeV A^b_FB = 0.1221 +- 0.0123 +- 0.0025 at sqrt(s) = 92.91 GeV where the first error is statistical and the second systematic in each case. Within the framework of the Standard Model, the result is interpreted as a measurement of the effective weak mixing angle for electrons of sin2theta_W^eff,e = 0.23205 +- 0.00068.