  Control over all internal and external degrees of freedom of molecules at the
level of single quantum states will enable a series of fundamental studies in
physics and chemistry. In particular, samples of ground-state molecules at
ultralow temperatures and high number densities will allow novel quantum-gas
studies and future applications in quantum information science. However, high
phase-space densities for molecular samples are not readily attainable as
efficient cooling techniques such as laser cooling are lacking. Here we produce
an ultracold and dense sample of molecules in a single hyperfine level of the
rovibronic ground state with each molecule individually trapped in the motional
ground state of an optical lattice well. Starting from a zero-temperature
atomic Mott-insulator state with optimized double-site occupancy, weakly-bound
dimer molecules are efficiently associated on a Feshbach resonance and
subsequently transferred to the rovibronic ground state by a stimulated
four-photon process with >50 % efficiency. The molecules are trapped in the
lattice and have a lifetime of 8 s. Our results present a crucial step towards
Bose-Einstein condensation of ground-state molecules and, when suitably
generalized to polar heteronuclear molecules, the realization of dipolar
quantum-gas phases in optical lattices.
