A lattice QCD calculation of the B meson decay constant is presented. In order to investigate the scaling violation associated with the heavy quark, parallel simulations are carried out employing both Wilson and the O(a)-improved clover actions for the heavy quark. The discretization errors due to the large b quark mass are estimated in a systematic way with the aid of the non-relativistic interpretation approach of El-Khadra, Kronfeld and Mackenzie. As our best value from the quenched simulations at beta=5.9,6.1 and 6.3 we obtain fB=163±16 MeV and fBs=175±18 MeV in the continuum limit where the error includes both statistical and systematic uncertainties.