  We solve the three-body problem of an ultracold Fermi gas with parabolic
confinement length $a_\perp$ and 3D scattering length $a$. On the two-body
level, there is a Feshbach-type resonance at $a_\perp/a\approx 1.46$, and a
dimer state for arbitrary $a_\perp/a$. The three-body problem is shown to be
universal, an d described by the atom-dimer scattering length $a_{ad}$ and a
range parameter $b_{ad}$. In the dimer limit $a_\perp/a\gg 1$, we find a
repulsive zero-range atom-dimer interaction. For $a_\perp/a\ll -1$, however,
the potential has long range, with $a_{ad}>0$ and $b_{ad}\gg a_{ad}$. There is
no trimer state, and despite $a_{ad}=0$ at $a_\perp/a\approx 2.6$, there is no
resonance enhancement of the interaction.
