  Let C be an ACM (projectively normal) nondegenerate smooth curve in
projective 3-space, and suppose C is general in its Hilbert scheme - this is
irreducible once the postulation is fixed. Answering a question posed by
Peskine, we show the gonality of C is d-l, where d is the degree of the curve,
and l is the maximum order of a multisecant line of C. Furthermore l=4 except
for two series of cases, in which the postulation of C forces every surface of
minimum degree containing C to contain a line as well. We compute the value of
l in terms of the postulation of C in these exceptional cases. We also show the
Clifford index of C is equal to the gonality minus 2.
