  We study effective energies of heavy-light meson correlation functions in
lattice QCD and a small volume of (0.2 fm)^4 to non-perturbatively calculate
their dependence on the heavy quark mass in the continuum limit. Our quenched
results obtained here constitute an essential intermediate step of a first
fully non-perturbative computation of the b-quark's mass in the static
approximation that has recently been presented as an application of a new
proposal to non-perturbatively renormalize the Heavy Quark Effective Theory.
The renormalization constant and the improvement coefficients relating the
renormalized current and subtracted quark mass are determined in the relevant
parameter region at weak couplings, which allows to perform the numerical
simulations at several, precisely fixed values of the renormalization group
invariant heavy quark mass in a range from 3 GeV to 15 GeV.
