  For any nonzero $h\in\mathbb{Z}$, we prove that a positive proportion of
integral binary cubic forms $F$ do locally everywhere represent $h$ but do not
globally represent $h$; that is, a positive proportion of cubic Thue equations
$F(x,y)=h$ fail the integral Hasse principle. Here, we order all classes of
such integral binary cubic forms $F$ by their absolute discriminants. We prove
the same result for Thue equations $G(x,y)=h$ of any fixed degree $n \geq 3$,
provided that these integral binary $n$-ic forms $G$ are ordered by the maximum
of the absolute values of their coefficients.
