The ratio of the CKM quark-mixing matrix elements |Vub|/|Vcb| has been measured using B hadron semileptonic decays. The analysis uses the reconstructed mass M_X of the secondary hadronic system produced in association with an identified lepton. Since B -> X_u lepton anti-nu transitions are characterised by hadronic masses below those of the D mesons produced in B -> X_c lepton anti-nu transitions, events with a reconstructed value of M_X significantly below the D mass are selected. Further signal enrichments are obtained using the topology of reconstructed decays and hadron identification. A fit to the numbers of decays in the b -> u enriched and depleted samples with M_X above and below 1.6 GeV/c^2 and to the shapes of the lepton energy distribution in the B rest frame gives |Vub|/|Vcb|= 0.103^{+0.011}_{-0.012} (stat.) +/- 0.016 (syst.) +/- 0.010 (model) and, correspondingly, a charmless semileptonic B decay branching fraction of BR(B -> X_u lepton anti-nu) = (1.57 +/- 0.35 (stat.) +/- 0.48 (syst.) +/- 0.27 (model)) x 10^{-3}.