  We consider a model describing the one-dimensional confinement of an exciton
in a symmetrical, rectangular quantum-well structure and derive upper and lower
bounds for the binding energy $E_b$ of the exciton. Based on these bounds, we
study the dependence of $E_b$ on the width of the confining potential with a
higher accuracy than previous reports. For an infinitely deep potential the
binding energy varies as expected from $1 Ry$ at large widths to $4 Ry$ at
small widths. For a finite potential, but without consideration of a mass
mismatch or a dielectric mismatch, we substantiate earlier results that the
binding energy approaches the value $1 Ry$ for both small and large widths,
having a characteristic peak for some intermediate size of the slab. Taking the
mismatch into account, this result will in general no longer be true. For the
specific case of a $Ga_{1-x}Al_{x}As/GaAs/Ga_{1-x}Al_{x}As$ quantum-well
structure, however, and in contrast to previous findings, the peak structure is
shown to survive.
