The decay B* --> B gamma has been observed with the DELPHI detector at LEP, where the B* meson 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.) MeV/c(2). 95% confidence level upper limits at 6 MeV/c(2) are placed on both the isospin (i.e, B+-B-0) and the B-s-B-ud 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 [x(E)] = 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 sigma(L)/(sigma(L) + sigma(T)) = 0.32 +/- 0.04 (stat.) +/- 0.03 (syst.).