  A novel high precision method measures the b-quark forward-backward asymmetry
at the Z pole on a sample of 3,560,890 hadronic events collected with the
DELPHI detector in 1992 to 2000. An enhanced impact parameter tag provides a
high purity b sample. For event hemispheres with a reconstructed secondary
vertex the charge of the corresponding quark or anti-quark is determined using
a neural network which combines in an optimal way the full available charge
information from the vertex charge, the jet charge and from identified leptons
and hadrons. The probability of correctly identifying b-quarks and anti-quarks
is measured on the data themselves comparing the rates of double hemisphere
tagged like-sign and unlike-sign events. The b-quark forward-backward asymmetry
is determined from the differential asymmetry, taking small corrections due to
hemisphere correlations and background contributions into account. The results
for different centre-of-mass energies are:
  A_{FB}^b (89.449 GeV) = 0.0637 \pm 0.0143(stat.) \pm 0.0017(syst.)
  A_{FB}^b (91.231 GeV) = 0.0958 \pm 0.0032(stat.) \pm 0.0014(syst.)
  A_{FB}^b (92.990 GeV) = 0.1041 \pm 0.0115(stat.) \pm 0.0024(syst.)
  Combining these results yields the b-quark pole asymmetry A_{FB}^{b,0} =
0.0972 \pm 0.0030(stat.) \pm 0.0014(syst.)
