  The absorption of \pi^+ on ^3He in the $\Delta$-region is evaluated with
exact inclusion of the final state interaction among the three emerging
protons. The absorption is described by a $\pi N \to \Delta$ vertex and a
$N\Delta - NN$ transition t-matrix which are calculated from a phenomenological
model for NN and \pi d reactions. In a calculation where the initial pion
scattering effects are neglected, the predicted peaks of the pion absorption
cross sections for ^2H and ^3He lie too high in energy in relation to the data.
The effect of the final state three-nucleon interaction turns out to be too
small for changing the magnitude and shifting the peak position of the total
absorption cross section for ^3He. We demonstrate that the adjustment of the
peak position for the deuteron cross section by small modifications of the
$\Delta$-parameters, automatically leads to the correct peak position in ^3He.
