  We present a method to measure the decay rate of the first excited
vibrational state of simple polar molecular ions being part of a Coulomb
crystal in a cryogenic linear Paul trap. Specifically, we have monitored the
decay of the $|\nu$=$1,J$=$1 \rangle_X$ towards the $|\nu$=$0,J$=$0 \rangle_X$
level in MgH$^+$ by saturated laser excitation of the $|\nu$=$0,J$=$2
\rangle_X$-$|\nu$=$1,J$=$1 \rangle_X$ transition followed by state selective
resonance enhanced two-photon dissociation out of the $|\nu$=$0,J$=$2
\rangle_X$ level. The technique enables the determination of decay rates, and
thus absorption strengths, with an accuracy at the few percent level.
