Résumé : Experimental evidence for the existence of orbitally excited B meson states is presented in an analysis of the Bπ and B*π distribution of Q = m(B**) - m(B(*)) - m(π) using Z0 decay data taken with the DELPHI detector at LEP. The mean Q-value of the decays B** → B(*)π is measured to be 284 ± 5 (stat.) ± 15 (syst.) MeV/c2, and the Gaussian width of the signal is 79 ± 5 (stat.) ± 8 (syst.) MeV/c2. This signal can be described as a single resonance of mass m = 5732 ± 5 (stat.) ± 20 (syst.) MeV/c2 and full width Γ = 145 ± 28 MeV/c2. The observed shape is also consistent with the production of several broad and narrow states as predicted by the quark model and partly observed in the D-meson sector. The production rate of B** per b-jet is found to be 0.27 ± 0.02 (stat.) ± 0.06 (syst.).