  We demonstrate a double-trap system well suited to study cold collisions
between trapped ions and trapped atoms. Using Yb$^+$ ions confined in a Paul
trap and Yb atoms in a magneto-optical trap, we investigate charge-exchange
collisions of several isotopes for collision energies down to 400 neV (5 mK).
The measured rate coefficient of $6 \times 10^{-10}$ cm$^{3}$s$^{-1}$, constant
over four orders of magnitude in collision energy, is in good agreement with
that derived from a semiclassical Langevin model for an atomic polarizability
of 143 a.u.
