The decay B* -t B¡ has been observed with the DELPHI detector at LEP, where the B* mesan is produced in Z boson decays. The combination of inclusively reconstructed B mesons with well-measured converted photons yields a measurement of the flavour-averaged B* - B mass difference of 45.5 ± 0.3 (stat.) ± 0.8 (syst.) Me V /c2 95% confidence leve! upper limits at 6 Me V /c2 are placed on both the isospin (i.e. B+ -Bº) and the B.-Bud splitting of the mass difference. The production ratio of B* to B mesons in Z decays is measured to be 0.72 ± 0.03 (stat.) ±0.06 (syst.). Limits on the production cross-section of other hypothetical excited B hadron states decaying radiatively are established. The differential B* cross section has been measured to be in good agreement with the average b fragmentation, yielding an average fractional B* energy of (xE) =0.695 ± 0.009 (stat.) ± 0.013 (syst.) . From the decay angular distribution the relative contribution of longitudinal B* polarisation states is measured to be <7L/(<7L + 17T) = 0.32 ± 0.04 (stat.) ±0.03 (syst.).