Curves on surfaces

Let $S$ be a smooth projective rational surface over an algebraically closed field of characteristic zero, and $Csubset X$ an irreducible curve with $C^2 = 0$.

Consider the linear system $|mathcal{O}_S(C)|$ of $C$. In this situation can we say that if $pin S$ is a general point then there exists at most one curve $Gammain |mathcal{O}_S(C)|$ passing through $p$?