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We demonstrate co-trapping and sideband cooling of a H$_2^+$ - $^9$Be$^+$ ionpair in a cryogenic Paul trap. We study the chemical lifetime of H$_2^+$ andits 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 oftranslational 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 ofH$_2^+$, as well as other light ions such as HD$^+$, H$_3^+$, and He$^+$.

Trapping and Ground-State Cooling of a Single H$_2^+$