Conventional information processors convert information between different physical carriers for processing, storage and transmission. It seems plausible that quantum information will also be held by different physical carriers in applications such as tests of fundamental physics, quantum enhanced sensors and quantum information processing. Quantum controlled molecules, in particular, could transduce quantum information across a wide range of quantum bit (qubit) frequencies-from a few kilohertz for transitions within the same rotational manifold(1), a few gigahertz for hyperfine transitions, a few terahertz for rotational transitions, to hundreds of terahertz for fundamental and overtone vibrational and electronic transitions-possibly all within the same molecule. Here we demonstrate entanglement between the rotational states of a (CaH+)-Ca-40 molecular ion and the internal states of a Ca-40(+) atomic ion(2). We extend methods used in quantum logic spectroscopy(1,3) for pure-state initialization, laser manipulation and state readout of the molecular ion. The quantum coherence of the Coulomb coupled motion between the atomic and molecular ions enables subsequent entangling manipulations. The qubit addressed in the molecule has a frequency of either 13.4 kilohertz(1) or 855 gigahertz(3), highlighting the versatility of molecular qubits. Our work demonstrates how molecules can transduce quantum information between qubits with different frequencies to enable hybrid quantum systems. We anticipate that our method of quantum control and measurement of molecules will find applications in quantum information science, quantum sensors, fundamental and applied physics, and controlled quantum chemistry. Quantum entanglement is realized between rotational levels of a molecular ion with energy differences spanning several orders of magnitude and long-lived internal states of a single atomic ion.