We investigate gravitational lensing using a realistic model of disc g
alaxies. Most of the mass is contained in a large spherical isothermal
dark matter halo, but the potential is modified significantly in the
core by a gravitationally dominant exponential disc. The method used i
s adapted from a very general multilens ray-tracing technique develope
d by Moller. We investigate the effects of the disc-to-halo mass ratio
, the disc scalelength, the disc inclination to the line of sight and
the lens redshift on two strong-lensing cross-sections: the cross-sect
ion for multiple imaging and the cross-section for large magnification
s, in excess of a factor of 10. We find that the multiple-imaging cros
s-section can be enhanced significantly by an almost edge-on Milky Way
disc compared with a singular isothermal sphere (SIS) in individual c
ases; however, when averaged over all disc inclinations, the cross-sec
tion is only increased by about 50 per cent. These results are consist
ent with other recent work. The presence of a disc, however, increases
the inclination-averaged high-magnification cross-section by an order
of magnitude compared with a SIS. This result has important implicati
ons for magnification bias in future lens surveys, particularly those
in the submillimetre waveband, where dust extinction in the lensing ga
laxy has no effect on the brightness of the images.