In hadronic decays of Z bosons recorded with the OPAL detector at LEP, events containing b quarks were selected using the long lifetime of b flavoured hadrons. Comparing the 3-jet rate in b events with that in d u,s and c quark events, a significant difference was observed. Using Order(alpha_s squared) calculations for massive quarks, this difference was used to determine the b quark mass in the MSbar renormalisation scheme at the scale of the Z boson mass. By combining the results from seven different jet finders the running b quark mass was determined to be mb(MZ) = (2.67 +/- 0.03(stat) +0.29/-0.37(syst) +/- 0.19(theo.)) GeV. Evolving this value to the b quark mass scale itself yields mb(mb) = (3.95 +0.52/-0.62) GeV, consistent with results obtained at the b quark production threshold. This determination confirms the QCD expectation of a scale dependent quark mass. A constant mass is ruled out by 3.9 standard deviations
