Recently, the quest for an ultracold and dense ensemble of polar molecules has attracted strong interest. Polar molecules have bright prospects for novel quantum gases with long-range and anisotropic interactions, for quantum information science, and for precision measurements. However, high-density clouds of ultracold polar molecules have so far not been produced. Here, we report a key step towards this goal. Starting from an ultracold dense gas of heteronuclear 40K-87Rb Feshbach molecules with typical binding energies of a few hundred kHz and a negligible dipole moment, we coherently transfer these molecules into a vibrational level of the ground-state molecular potential bound by >10 GHz. We thereby increase the binding energy and the expected dipole moment of the 40K-87Rb molecules by more than four orders of magnitude in a single transfer step. Starting with a single initial state prepared with Feshbach association, we achieve a transfer efficiency of 84%. While dipolar effects are not yet observable, the presented technique can be extended to access much more deeply bound vibrational levels and ultimately those exhibiting a significant dipole moment. The preparation of an ultracold quantum gas of polar molecules might therefore come within experimental reach.