  In 1986, Zamolodchikov conjectured an exponential structure for the
semi-classical limit of conformal blocks on a sphere. This paper provides a
rigorous proof of the analog of Zamolodchikov conjecture for Liouville
conformal blocks on a one-punctured torus, using their probabilistic
construction and show the existence of a positive radius of convergence of the
semi-classical limit. As a consequence, we obtain a closed form expression for
the solution of the Lam\'e equation, and show a relation between its accessory
parameter and the classical action of the non-autonomous elliptic
Calogero-Moser model evaluated at specific values of the solution.
