  We analyze reactive collisions of a single Ba$^+$ ion in contact with an
ultracold gas of Rb atoms at low three-body collision energies of 2.2(9)
mK$\times k_{\mathrm{B}}$. Mapping out the Ba$^+$ loss rate dependence on the
Rb atom density we can discern two-body reactive collisions from three-body
ones and for the first time determine both rate coefficients which are
$k_2=3.1(6)(6)\times 10^{-13}\textrm{cm}^{3}\textrm{s}^{-1}$ and
$k_3=1.04(4)(45)\times 10^{-24}\textrm{cm}^{6}\textrm{s}^{-1}$, respectively
(statistical and systematic errors in parenthesis). Thus, the measured ternary
recombination dominates over binary reactions even at moderate atom densities
of $n\approx 10^{12}\: \textrm{cm}^{-3}$. The results for Ba$^+$ and Rb are
representative for a wide range of cold ion-atom systems and can serve as a
guidance for the future development of the field of hybrid atom-ion research.
