From a data sample of 450 000 hadronic events recorded with the ALEPH detector at LEP, 92±10 events are observed containing a J/ψ meson decaying to μ+μ− or e+e−. From these data the measured inclusive branching fraction for a b flavoured hadron to decay to a J/ψ is BR(b→ J/ψX) = (1.21±0.13 (stat.)±0.08 (syst.))%, and the average b hadron lifetime in the events tagged with a J/ψ is τb = 1.35+0.19−0.17±0.05 ps. Five events are observed consistent with the exclusive decay B± → J/ψK± and from these events the exclusive branching fraction is measured to be BR(B± → J/ψK±) = (0.22±0.10±0.02)%. Upper limits for other exclusive branching ratios are given.