La4Cu3MoO12 is a new (ABO(3))(n=4) cuprate with mixed B-cations in a ratio of 1:3. When synthesized at ambient pressure, the structure is not perovskite as expected but rather a homeotype of YAlO3, a rare-earth hexagonal phase. While the P6(3)/mmc and Pmnm space groups can be used to model average structures which appear in quenched samples; powder, electron, and neutron diffraction data all confirm that a slow-cooled sample crystallizes in the monoclinic space group, P112(1)/m. The copper and molybdenum are coordinated by oxygen in corner-sharing trigonal bipyramids that are sandwiched between layers of lanthanum cations. In the B-cation layer, the copper cations order into a kagome-like lattice of triangular clusters. The magnetism has been measured from 2 to 800 K and is highly influenced by the geometric arrangement of the Cu-II cations. An antiferromagnetic transition occurs at 5 K, but the sample does not reach a purely paramagnetic state until 460 K.