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We have established the strontium molecular clock as the leading ultracold-molecule precision measurement platform. We can fully control the molecules internal and external quantum states, as in state-of-the-art atomic lattice clocks. Using the quantized molecular spectra in an optical lattice, we devised a new lattice thermometry method that is advantageous for many lattice experiments, including clocks and manybody physics studies. The full quantum control allowed us to observe deeply subradiant molecular states, and coherently control them. We measured lifetimes enhanced by up to 300 times. Ultranarrow transitions to these states permitted a rigorous study of asymptotic physics at the atom-molecule threshold. We proved that subradiant states can decay to the ground state via higher-order electromagnetic transitions, with a rate that grows as the square of the bond length, and are quenched by predissociation at a rate proportional to the vibrational level spacing. We introduced a precise technique for measuring molecular binding energies via photofragmentation; controlled forbidden optical transition strengths with weak magnetic fields by five orders of magnitude; and explained anomalously large magnetic susceptibilities of the asymptotic molecules. The unprecedented precision, coupled with advanced ab initio theory, paves the way to exciting new directions for the molecular clock.