  We apply solutions of Heun's general equation to the stationary Schr\"odinger
equation with two quasi-exactly solvable elliptic potentials which depend on a
real parameter $\ell$. We get finite-series solutions from power series
expansions for Heun's equation if $\ell$ is an integer, except if
$\ell=-1,-2,-3,-4$. If $\ell\neq-5/2$ is half an odd integer, we obtain finite
series in terms of hypergeometric functions. The quasi-exact solvability is
expressed by the finite series solutions. However, for any value of $\ell$, we
find infinite-series eigenfunctions which are convergent and bounded for all
values of the independent variable.
