  We describe a method for cooling neutral molecules that have magnetic and
electric dipole moments using collisions with cold ions. An external magnetic
field is used to split the ground rovibrational energy levels of the molecule.
The highest energy state within the ground rovibrational manifold increases in
energy as the distance to the ion decreases leading to a repelling potential.
At low energy, inelastic collisions are strongly suppressed due to the large
distance of closest approach. Thus, a collision between a neutral molecule and
a cold ion will lead to a decrease in the molecule's kinetic energy with no
change in internal energy. We present results for the specific case of OH
molecules cooled by Be$^+$, Mg$^+$, or Ca$^+$ ions.
