We compute decay constants of heavy–light mesons in quenched lattice QCD with a lattice spacing of asimilar, equals0.04 fm using non-perturbatively O(a) improved Wilson fermions and O(a) improved currents. We obtain fDs=220(6)(5)(11) MeV, fD=206(6)(3)(22) MeV, fBs=205(7)(26)(17) MeV and fB=190(8)(23)(25) MeV, using the Sommer parameter r0=0.5 fm to set the scale. The first error is statistical, the second systematic and the third from assuming a ±10% uncertainty in the experimental value of r0. A detailed discussion is given in the text. We also present results for the meson decay constants fK and fπ and the ρ meson mass.