We prove the existence of a set of two-scale magnetic Wannier orbitals, wmn(r), in the infinite plane. The quantum numbers of these states are the positions (m,n) of their centers which form a von Neumann lattice. Function w00(r) localized at the origin has a nearly Gaussian shape of exp(-r2/4l2)/2π for r<~2πl, where l is the magnetic length. This region makes a dominating contribution to the normalization integral. Outside this region function w00(r) is small, oscillates, and falls off with the Thouless critical exponent for magnetic orbitals, r-2. These functions form a complete basis for many-electron problems.