  Let $m\geq -1$ be an integer. We give a correspondence between integer
solutions to the parametric family of cubic Thue equations \[
X^3-mX^2Y-(m+3)XY^2-Y^3=\lambda \] where $\lambda>0$ is a divisor of $m^2+3m+9$
and isomorphism classes of the simplest cubic fields. By the correspondence and
R. Okazaki's result, we determine the exactly 66 non-trivial solutions to the
Thue equations for positive divisors $\lambda$ of $m^2+3m+9$. As a consequence,
we obtain another proof of Okazaki's theorem which asserts that the simplest
cubic fields are non-isomorphic to each other except for
$m=-1,0,1,2,3,5,12,54,66,1259,2389$.
