The unitary representations of the affine group, or the group of linear transformations without reflections on the real line, have been found previously by Gel'fand and Naimark. The present paper gives an alternate proof, and presents several properties of the representations which will be used in a later application of this group to continuous representations of Hilbert space. The development follows closely that used by von Neumann to prove the uniqueness of the Schrödinger operators.