We use a quantum-logic spectroscopy (QLS) protocol to control the quantum state of a ${\mathrm{CaH}}^{+}$ ion in a cryogenic environment in which reduced thermal radiation extends rotational state lifetimes by an order of magnitude over those at room temperature. By repeatedly and adaptively probing the molecule, detecting the outcome of each probe via an atomic ion, and using a Bayesian update scheme to quantify confidence in the molecular state, we demonstrate state preparation and measurement (SPAM) in a single quantum state with infidelity less than $6\times {10}^{-3}$ and measure Rabi flopping between two states with greater than 99% contrast. The protocol does not require molecule-specific lasers and can be generalized, paving the way for high-fidelity quantum control of many other molecular ions.

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• Quantum control
• Quantum measurements
• Ionized molecules
• Trapped ions