We report the successful production of subradiant states of a two-atom system in a three-dimensional optical lattice starting from doubly occupied sites in a Mott insulator phase of a quantum gas of atomic ytterbium. We can selectively produce either a subradiant 1(g) state or a superradiant 0(u) state by choosing the excitation laser frequency. The inherent weak excitation rate for the subradiant 1(g) state is overcome by the increased atomic density due to the tight confinement in a three-dimensional optical lattice. Our experimental measurements of binding energies, linewidth, and Zeeman shift confirm the observation of subradiant levels of the 1(g) state of the Yb-2 molecule.