The ratio of the CKM quark-mixing matrix elements |V ub |/|V cb | 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 ℓν transitions are characterised by hadronic masses below those of the D mesons produced in B → X c ℓν 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 |V ub |/|V cb | = 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 ℓν) = (1.57 ± 0.35 (stat.) ± 0.48 (syst.) ± 0.27 (model)) × 10 −3 .