  We report on the production of cold, state-selected H$_2^+$ molecular ions in
a linear RF trap. The ions are produced by (3+1) resonance-enhanced
multi-photon ionisation (REMPI) of H$_2$, and sympathetically cooled by
laser-cooled Be$^+$ ions. After demonstrating and characterizing the REMPI
process, we use photodissociation by a deep UV laser at 213~nm to verify the
high vibrational purity of the produced H$_2^+$ ion samples. Moreover, the
large difference between the photodissociation efficiencies of ions created in
the $v=0$ and $v=1$ levels provides a way to detect a $v=0 \to 1$ transition.
These results pave the way towards high-resolution vibrational spectroscopy of
H$_2^+$ for fundamental metrology applications.
