On Legendrian curves in ${\mathbb{P}}^3$

We show that if a smooth projective curve $C\subset {\mathbb{P}}^3$ (over an algebraically closed field of characteristic zero) is Legendrian with respect to a contact structure (it is well known that a contact structure on $\mathbb P^3$isuniqueuptoalinearautomorphism)and$C$ is linearly normal (i.e., not an isomorphic linear projection of a smooth curve $C^{\mathrm{\prime }}\subset {\mathbb{P}}^n$, $n>3$, where $C^{\mathrm{\prime }}$ does not lie in a hyperplane) then $C$ is a twisted cubic or a line.