This is joint work with R. Hartshorne. Let C be an ACM (projectively
normal) nonsingular curve in P^3 not contained in a plane, and suppose C
is general in its Hilbert scheme - this is irreducible once the
postulation is fixed.

Answering a question by Peskine, we show the gonality of C is d-m, where d
is the degree of the curve, and m is the maximum order of a multisecant to
C.  Furthermore m=4 except for rare cases, when the postulation of C
forces every surface of minumum degree containing C to contain a line as
well. We compute the value of m in these exceptional cases as well.