  We present a procedure for computing gauge-invariant scattering amplitudes in
the $W_3$ string, and use it to calculate three-point and four-point functions.
We show that non-vanishing scattering amplitudes necessarily involve external
physical states with excitations of ghosts as well as matter fields. The
crossing properties of the four-point functions are studied, and it is shown
that the duality of the Virasoro string amplitudes generalises in the $W_3$
string, with different sets of intermediate states being exchanged in different
channels. We also exhibit a relation between the scattering amplitudes of the
$W_3$ string and the fusion rules of the Ising model.
