Precision laser spectroscopy 1 of cold and trapped molecular ions is a powerful tool for fundamental physics, including the determination of fundamental constants 2 , the laboratory test for their possible variation 3,4 , and the search for a possible electric dipole moment of the electron 5 . While the complexity of molecular structure facilitates these applications, the absence of cycling transitions poses a challenge for direct laser cooling 6 , quantum state control [7][8][9][10][11] , and detection. Previously employed state detection techniques based on photodissociation 12 or chemical reactions 13 are destructive and therefore inefficient, restricting the achievable resolution in laser spectroscopy. Here we experimentally demonstrate nondestructive state detection of a single trapped molecular ion through its strong Coulomb coupling to a well-controlled co-trapped atomic ion. An algorithm based on a state-dependent optical dipole force 14 (ODF) changes the internal state of the atom conditioned on the internal state of the molecule. We show that individual quantum states in the molecular ion can be distinguished by their coupling strength to the ODF and observe black-body radiationinduced quantum jumps between rotational states of a single molecular ion. Using the detuning dependence of the state detection signal, we implement a variant of quantum logic spectroscopy 15,16 of a molecular resonance. The state detection technique we demonstrate is applicable to a wide range of molecular ions, enabling further applications in state-controlled quantum chemistry 17 and spectroscopic investigations of molecules serving as probes for interstellar clouds 18,19 .One of the salient features of trapped ion systems is that the universal Coulomb interaction allows strong coupling of diverse quantum objects, such as different species of atomic ions or atomic and molecular ions. Being able to perform quantum logic operations e.g. in the form of gates 14,20,21 between the quantum objects has proven a powerful tool for quantum information processing and quantum simulations in such systems. It also allows combining the advantages of different atomic species. Quantum logic spectroscopy is one such application in which the high degree of control achieved over selected atomic ions is extended to species over which such control is lacking 15,16 . Here, we demonstrate for the first time quantum logic operations between a single molecular ion and a co-trapped atomic ion, making a wide range of molecular ions accessible to this highlydeveloped toolbox. The presented technique allows the investigation of single molecules in a well isolated system avoiding disturbance from the environment, which is the limiting factor in other implementations of single molecule spectroscopy such as surface enhanced Raman spectroscopy (SERS) 22 Quantum logic operations between atoms are based on state dependent forces often induced by laser fields. The same approach is applicable to molecular ions. The coupling is now distributed over many ro-vibrat...