  We construct B\"acklund transformations (BTs) for the Kirchhoff top by taking
advantage of the common algebraic Poisson structure between this system and the
$sl(2)$ trigonometric Gaudin model. Our BTs are integrable maps providing an
exact time-discretization of the system, inasmuch as they preserve both its
Poisson structure and its invariants. Moreover, in some special cases we are
able to show that these maps can be explicitly integrated in terms of the
initial conditions and of the "iteration time" $n$. Encouraged by these partial
results we make the conjecture that the maps are interpolated by a specific
one-parameter family of hamiltonian flows, and present the corresponding
solution. We enclose a few pictures where the orbits of the continuous and of
the discrete flow are depicted.
