We report the experimental realization of a conveyor-belt magneto-optical trap for calcium monofluoride (CaF) molecules. The obtained highly-compressed cloud has a mean radius of 64(5) µm and a peak number density of $3.6(5)\times 10^10$ cm$^{-3}$, a 600-fold increase over the conventional red-detuned MOTs of CaF, and the densest molecular MOT observed to date. Subsequent loading of these molecules into an optical dipole trap yields up to $2.6\times 10^4$ trapped molecules at a temperature of 14(2) µK with a peak phase-space density of $2.4\times 10^{-6}$. This opens new possibilities for a range of applications utilizing high-density, optically trapped ultracold molecules.