  I show that the potential $$V(x,m) = \big [\frac{b^2}{4}-m(1-m)a(a+1) \big
]\frac{\sn^2 (x,m)}{\dn^2 (x,m)} -b(a+{1/2}) \frac{\cn (x,m)}{\dn^2 (x,m)}$$
constitutes a QES band-structure problem in one dimension. In particular, I
show that for any positive integral or half-integral $a$, $2a+1$ band edge
eigenvalues and eigenfunctions can be obtained analytically. In the limit of m
going to 0 or 1, I recover the well known results for the QES double
sine-Gordon or double sinh-Gordon equations respectively. As a by product, I
also obtain the boundstate eigenvalues and eigenfunctions of the potential
$$V(x) = \big [\frac{\beta^2}{4}-a(a+1) \big ] \sech^2 x +\beta(a+{1/2})\sech
x\tanh x$$ in case $a$ is any positive integer or half-integer.
