  The study of polynomial solutions to the classical Lam\'e equation in its
algebraic form, or equivalently, of double-periodic solutions of its
Weierstrass form has a long history. Such solutions appear at integer values of
the spectral parameter and their respective eigenvalues serve as the ends of
bands in the boundary value problem for the corresponding Schr\"odinger
equation with finite gap potential given by the Weierstrass $\wp$-function on
the real line. In this paper we establish several natural (and equivalent)
formulas in terms of hypergeometric and elliptic type integrals for the density
of the appropriately scaled asymptotic distribution of these eigenvalues when
the integer-valued spectral parameter tends to infinity. We also show that this
density satisfies a Heun differential equation with four singularities.
