  In weakly bound diatomic molecules, energy levels are closely spaced and thus
more susceptible to mixing by magnetic fields than in the constituent atoms. We
use this effect to control the strengths of forbidden optical transitions in
$^{88}$Sr$_2$ over 5 orders of magnitude with modest fields by taking advantage
of the intercombination-line threshold. The physics behind this remarkable
tunability is accurately explained with both a simple model and quantum
chemistry calculations, and suggests new possibilities for molecular clocks. We
show how mixed quantization in an optical lattice can simplify molecular
spectroscopy. Furthermore, our observation of formerly inaccessible $f$-parity
excited states offers an avenue for improving theoretical models of
divalent-atom dimers.
