We present the results of a numerical calculation of semi-leptonic form factors relevant for heavy flavour meson decays into light mesons, at β=6.4 on a 243×60 lattice, using the Wilson action in the quenched approximation. We obtain f+K(0)=0.65±0.18, V(0)=0.95±0.34, A1(0)=0.63±0.14 and A2(0)=0.45±0.33. We also obtain A1(q2max)=0.62±0.09, V(0)/A1(0)=1.5±0.28 and A2(0)/A1(0)=0.7±0.4. The results for f+K(0), V(0) and A1(0) are consistent with the experimental data and with previous lattice determinations with larger lattice spacings. In the case of A2(0) the errors are too large to draw any firm conclusion. We have also extrapolated the form factors to the B meson, showing a behaviour compatible with the predictions by the heavy quark effective theory (HQET). Within large uncertainties, our results suggest that A2/A1 increases with the heavy quark mass. We also get very rough estimates for the partial decay widths B→πlνl)=|Vub|2(12±8)1012s−1 and Γ(B→ρlνl)=|Vub|2(13±12)1012s−1, which can be used to give upper bounds on the rates.