  Entangled quantum states share properties that do not have classical analogs,
in particular, they show correlations that can violate Bell inequalities. It is
therefore an interesting question to see what happens to entanglement measures
-- such as the entanglement entropy for a pure state -- taking the
semi-classical limit, where the naive expectation is that they may become
singular or zero. This conclusion is however incorrect. In this paper, we
determine the $\hbar\rightarrow 0$ limit of the bipartite entanglement entropy
for a one-dimensional system of $N$ quantum particles in an external potential
and we explicitly show that this limit is finite. Moreover, if the particles
are fermionic, we show that the $\hbar\rightarrow 0$ limit of the bipartite
entanglement entropy coincides with the Shannon entropy of $N$ bits.
