We present a high statistics study of the D- and B-meson decay constants. The results were obtained by using the Clover and Wilson lattice actions at two different values of the lattice spacing a, corresponding to beta = 6.0 and 6.2. After a careful analysis of the systematic errors present in the extraction of the physical results, by assuming quite conservative discretization errors, we find f(Ds) = 237 +/- 16 MeV, f(D) = 221 +/- 17 MeV (f(Ds)/f(D) = 1.07(4)), f(Bs) = 205 +/- 35 MeV, f(B) = 180 +/- 32 MeV (f(Bs)/f(B) = 1.14(8)), in good agreement with previous estimates. (C) 1997 Elsevier Science B.V.