  Let ${\{K_m\}_{m\geq 4}}$ be the family of non-normal totally real cubic
number fields defined by the irreducible cubic polynomial
$f_m(x)=x^3-mx^2-(m+1)x-1$, where $m$ is an integer with $m\geq 4$. In this
paper, we will give a class number one criterion for $K_m$.
