The mean field like gauge invariant variational method formulated recently, is applied to a topologically massive QED in 3 dimensions. We find that the theory has a phase transition in the Chern Simons coefficient n. The phase transition is of the Berezinsky-Kosterlitz - Thouless type, and is triggered by the liberation of Polyakov monopoles, which for n>8 are tightly bound into pairs. In our Hamiltonian approach this is seen as a similar behaviour of the magnetic vortices, which are present in the ground state wave functional of the compact theory. For n>8, the low energy behavior of the theory is the same as in the noncompact case. For n<8 there are no propagating degrees of freedom on distance scales larger than the ultraviolet cutoff. The distinguishing property of the n<8 phase, is that the magnetic flux symmetry is spontaneoously broken
