A simple but rigorous solution of the infrared problem is obtained. The basis of this solution is a factorization of the operator corresponding to the Feynman coordinate space plus all electromagnetic corrections to it into a product of two operators. The first is a unitary operator that represents precisely the contribution corresponding to classical electromagnetic theory. The second is a residual operator that is free of infrared problems. This factorization is exact: no soft-photon approximation, or any other approximation, is used. Both the unitary operator and the residual operator are expressed in simple forms amenable to rigorous mathematical analysis. The central technical result of this work, namely the exact yet simple organization of all contributions corresponding to classical physics into unitary factors, may have other important uses.