Sr2Cu3O4Cl2 has CuI and CuII subsystems, forming interpenetrating S=1/2 square lattice Heisenberg antiferromagnets. The classical ground state is degenerate, due to frustration of the intersubsystem interactions. Magnetic neutron scattering experiments show that quantum fluctuations cause a two dimensional Ising ordering of the CuII's, lifting the degeneracy, and a dramatic increase of the CuI out-of-plane spin-wave gap, unique for order out of disorder. The spin-wave energies are quantitatively predicted by calculations which include quantum fluctuations.