  We theoretically study the dynamics of a trapped ion that is immersed in an
ultracold gas of weakly bound atomic dimers created by a Feshbach resonance.
Using quasi-classical simulations, we find a crossover from dimer dissociation
to molecular ion formation depending on the binding energy of the dimers. The
location of the crossover strongly depends on the collision energy and the
time-dependent fields of the Paul trap. Deeply bound dimers lead to fast
molecular ion formation, with rates approaching the Langevin collision rate
$\Gamma'_\text{L}\approx4.8\times10^{-9}\,$cm$^3$s$^{-1}$. The kinetic energies
of the created molecular ions have a median below $1\,$mK, such that they will
stay confined in the ion trap. We conclude that interacting ions and Feshbach
molecules may provide a novel approach towards the creation of ultracold
molecular ions with applications in precision spectroscopy and quantum
chemistry.
