  We demonstrate co-trapping and sideband cooling of a H$_2^+$ - $^9$Be$^+$ ion
pair in a cryogenic Paul trap. We study the chemical lifetime of H$_2^+$ and
its dependence on the apparatus temperature, achieving lifetimes of up to
$11^{+6}_{-3}$ h at 10 K. We demonstrate cooling of two of the modes of
translational motion to an average phonon number of 0.07(1) and 0.05(1),
corresponding to a temperature of 22(1) $\mu$K and 55(3) $\mu$K respectively.
Our results provide a basis for quantum logic spectroscopy experiments of
H$_2^+$, as well as other light ions such as HD$^+$, H$_3^+$, and He$^+$.
