The light-hole exciton in GaAs-Ga$_{1-x}$Al$_x$As quantum wells is degenerate with the continuum of the heavy-hole exciton for well widths L < 130 A. The two exciton series are coupled, due to valence-band mixing. Thus the light-hole exciton acquires a resonance broadening. In this work this broadening is calculated. We present two methods which are both based on the diagonalisation of the Hamiltonian on a nite variational basis, chosen in such a way that the discrete as well as the continuum parts of the spectrum are well represented.