  Sympathetic cooling of trapped ions has been established as a powerful
technique for manipulation of non-laser-coolable ions
(Raizen1992,Waki1992,Bowe1999,Barrett2003). For molecular ions, it promises
vastly enhanced spectroscopic resolution and accuracy. However, this potential
remains untapped so far, with the best resolution achieved being not better
than $5\times10^{-8}$ fractionally, due to residual Doppler broadening being
present in ion clusters even at the lowest achievable translational
temperatures (Bressel2012). Here we introduce a general and accessible approach
that enables Doppler-free rotational spectroscopy. It makes use of the strong
radial spatial confinement of molecular ions when trapped and crystallized in a
linear quadrupole trap, providing the Lamb-Dicke regime for rotational
transitions. We achieve a line width of $1\times10^{-9}$ fractionally and
$1.3~\textrm{kHz}$ absolute, an improvement by $50$ and nearly $3\times10^{3}$,
respectively, over other methods. The systematic uncertainty is
$2.5\times10^{-10}$. As an application, we demonstrate the most precise test of
$\textit{ab initio}$ molecular theory and the most precise ($1.3~\textrm{PPB}$)
spectroscopic determination of the proton mass. The results represent the long
overdue extension of Doppler-free microwave spectroscopy of laser-cooled atomic
ion clusters (Berkeland1998) to higher spectroscopy frequencies and to
molecules. This approach enables a vast range of high-precision measurements on
molecules, both on rotational and, as we project, vibrational transitions.
