In hadronic decays of Z bosons recorded with the OPAL detector at LEP, even
ts 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 qua
rk events, a significant difference was observed. Using O(alpha (2)(S)) cal
culations for massive quarks, this difference was used to determine the b q
uark mass in the YI-S renormalisation scheme at the scale of the Z boson ma
ss. By combining the results from seven different jet finders the running b
quark mass was determined to be (m) over bar (b) (m(Z)) = (2.67 +/-0.03(st
at.)(-0.37)(+0.29)(syst.)+/-0.19(theo.)) GeV. Evolving this value to the b
quark mass scale itself yields (m) over bar (b)((m) over bar (b)) = (3.95(-
0.62)(+0.52))GeV, consistent with results obtained at the b quark productio
n threshold. This determination confirms the QCD expectation of a scale dep
endent quark mass. A constant mass is ruled out by 3.9 standard deviations.