  The purpose of this paper is to show that for a complete intersection curve
$C$ in projective space (other than a few stated exceptions), any morphism $f:
C \to \mathbb{P}^r$ satisfying $\text{deg}\, f^*\mathcal{O}_{\mathbb{P}^r}(1)
<\text{deg}\, C$ is obtained by projection from a linear space. In particular,
we obtain bounds on the gonality of such curves and compute the gonality of
general complete intersection curves. We also prove a special case of one of
the well-known Cayley-Bacharach conjectures posed by Eisenbud, Green, and
Harris.
