  The nature of b-quark jet hadronisation has been investigated using data
taken at the Z peak by the DELPHI detector at LEP. Two complementary methods
are used to reconstruct the energy of weakly decaying b-hadrons, E^weak_B. The
average value of x^weak_B = E^weak_B/E_beam is measured to be 0.699 +/- 0.011.
The resulting x^weak_B distribution is then analysed in the framework of two
choices for the perturbative contribution (parton shower and Next to Leading
Log QCD calculation) in order to extract measurements of the non-perturbative
contribution to be used in studies of b-hadron production in other experimental
environments than LEP. In the parton shower framework, data favour the Lund
model ansatz and corresponding values of its parameters have been determined
within PYTHIA~6.156 from DELPHI data: a= 1.84^{+0.23}_{-0.21} and
b=0.642^{+0.073}_{-0.063} GeV^-2, with a correlation factor rho = 92.2%.
Combining the data on the b-quark fragmentation distributions with those
obtained at the Z peak by ALEPH, OPAL and SLD, the average value of x^weak_B is
found to be 0.7092 +/- 0.0025 and the non-perturbative fragmentation component
is extracted. Using the combined distribution, a better determination of the
Lund parameters is also obtained: a= 1.48^{+0.11}_{-0.10} and
b=0.509^{+0.024}_{-0.023} GeV^-2, with a correlation factor rho = 92.6%.
