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 O(α2S) calculations for massive quarks, this difference was used to determine the b quark mass in the MS¯ 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 m¯b(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 m¯b(m¯b)=(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.