Sympathetic cooling of trapped ions has been established as a powerful technique for manipulation of non-laser-coolable ions [1][2][3][4]. 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 × 10 −8 fractionally, due to residual Doppler broadening being present in ion clusters even at the lowest achievable translational temperatures [5]. 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 × 10 −9 fractionally and 1.3 kHz absolute, an improvement by 50 and nearly 3 × 10 3 , respectively, over other methods. The systematic uncertainty is 2.5 × 10 −10 . As an application, we demonstrate the most precise test of ab initio molecular theory and the most precise (1.3 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 [6] 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. arXiv:1802.03208v1 [quant-ph] 9 Feb 2018 pair were split and resolved, the Zeeman shift uncertainty should be reduced at least 10-fold.This would then allow a total systematic uncertainty of < 3 × 10 −11 . ACKNOWLEDGMENTS This work has been partially funded by DFG project Schi 431/21-1. We thank U. Rosowski for important assistance with the frequency comb, A. Nevsky for assistance with a laser system, E. Wiens for characterizing H-maser instability, R. Gusek and P. Dutkiewicz for electronics development, J. Scheuer and M. Melzer for assistance, and S. Schlemmer (Universität zu Köln) for equipment loans. We thank K. Brown (Georgia Institute of Technology) for useful discussions and suggestions. Corresponding author, step.schiller@hhu.de Contributions S.A. and M.G.H. developed the apparatus and performed the experiments, S.A., M.G.H., and S.S. analyzed the data, S.A., S.S. and V.I.K. performed theoretical calculations, S.S.conceived the study, supervised the work and wrote the paper.