We calculate the B-meson decay constant, f_B, and the form factors, f⁺(0), A₁(0) and V(0), of the decays B̅;⁰ → πev̅;_e and B̅;⁰ → ρev̅;_e in the limit that the b quark is static. The calculations are performed on a 24³ x 48 lattice at β = 6.2 in the quenched approximation. The different methods we use to obtain f_B give results between 237 (16) and 283 (25) MeV, in good agreement with static-limit calculations performed by other authors. The matrix elements of the decays mentioned are well-determined at the momentum transfers and the masses of the light quarks which we considered. This gives confidence to the application of the static-limit method to the study of the semi-leptonic decays of heavy-light mesons. However, while the extrapolation processes necessary for arriving at f⁺ (0), A₁(0) and V(0) give results similar to those obtained by the authors who use a method in which the b quark propagates, they also introduce large uncertainties, so that no definite statements can be made about these form factors.