  Maclagan and Smith \cite{MaclaganSmith} developed a multigraded version of
Castelnuovo-Mumford regularity. Based on their definition we will prove in this
paper that for a smooth curve $C\subseteq \P^a\times\P^b$ $(a, b\geq 2)$ of
bidegree $(d_1,d_2)$ with nondegenerate birational projections the ideal sheaf
$\mathcal{I}_{C|\P^a\times\P^b}$ is $(d_2-b+1,d_1-a+1)$-regular. We also give
an example showing that in some cases this bound is the best possible.
