  We compute decay constants of heavy-light mesons in quenched lattice QCD with
a lattice spacing of a ~ 0.04 fm using non-perturbatively O(a) improved Wilson
fermions and O(a) improved currents. We obtain f_{D_s} = 220(6)(5)(11) MeV, f_D
= 206(6)(3)(22) MeV, f_{B_s} = 205(7)(26)(17) MeV and f_B = 190(8)(23)(25) MeV,
using the Sommer parameter r_0 = 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 r_0. A detailed discussion is given in
the text. We also present results for the meson decay constants f_K and f_\pi
and the \rho meson mass.
