In this work, we demonstrate a new method for quantum state control and nondestructive internal quantum state readout of a single HD^{+} ion in the rovibrational ground state in a Penning trap. Furthermore, we demonstrate a measurement of the magnetic trapping field with the same ion to sub-ppb uncertainty. This could enable nondestructive, high-precision spectroscopy of the hyperfine, Zeeman, and rovibrational level structure of single molecular ions. In particular, this technique can be applied to H_{2}^{+} as well as to its antimatter equivalent, H[over ¯]_{2}^{-}, with the potential for high-precision tests of charge-parity-time reversal symmetry beyond current sensitivities [Phys. Scr. T 1995, 423 (1995)PHSTBO0031-894910.1088/0031-8949/1995/T59/060, Phys. Rev. A 98, 010101 (2018)PLRAAN2469-992610.1103/PhysRevA.98.010101].