  Optical spectroscopy in the gas phase is one of the key tools for the
elucidation of the structure of atoms and molecules and their interaction with
external fields. The line resolution is usually limited by a combination of
first-order Doppler broadening due to particle thermal motion and of a short
transit time through the excitation beam. For trapped particles, suitable laser
cooling techniques can lead to strong confinement (Lamb-Dicke regime, LDR) and
thus to optical spectroscopy free of these effects. For non-laser coolable
ions, this has so far only been achieved when trapping one or two atomic ions,
together with a single laser-coolable atomic ion. Here we show that one-photon
optical LDR spectroscopy can also be obtained with more easily prepared
ensembles of ions, if performed in the mid-infrared spectral range. We
demonstrate the method on molecular ions, for which LDR vibrational
spectroscopy has not before been achieved. We trap approximately 100 molecular
hydrogen ions (HD$^{+}$) within a Coulomb cluster of a few thousand
laser-cooled atomic ions and perform laser spectroscopy of the fundamental
vibrational transition. To generate the appropriate spectroscopy radiation (5.1
$\mu$m), a metrology-grade, mid-infrared laser source was developed. We obtain
fractional vibrational linewidths of $3\times10^{-12}$, $10^{4}$ times smaller
than previous best results for one-photon vibrational spectroscopy of molecular
ions. Transition frequencies were determined with lowest uncertainties
$3.3\times10^{-12}$ fractionally. As an application, we determine the
proton-electron mass ratio by matching a precise $ab~initio$ calculation with
the measured vibrational frequency. The mass ratio is in agreement with the
most precise recent measurements and has a similar uncertainty. The present
technique is relatively simple and applicable to a large variety of molecular
ions.
