  Assume $a$ and $b=na+r$ with $n \geq 1$ and $0<r<a$ are relatively prime
integers. In case $C$ is a smooth curve and $P$ is a point on $C$ with
Weierstrass semigroup equal to $<a;b>$ then $C$ is called a $C_{a;b}$-curve. In
case $r \neq a-1$ and $b \neq a+1$ we prove $C$ has no other point $Q \neq P$
having Weierstrass semigroup equal to $<a;b>$. We say the Weierstrass semigroup
$<a;b>$ occurs at most once. The curve $C_{a;b}$ has genus $(a-1)(b-1)/2$ and
the result is generalized to genus $g<(a-1)(b-1)/2$. We obtain a lower bound on
$g$ (sharp in many cases) such that all Weierstrass semigroups of genus $g$
containing $<a;b>$ occur at most once.
