A polarimeter was constructed to measure the longitudinal polarization of a spin-polarized electron beam at 5 and 7 MeV. The polarimeter takes advantage of Compton scattering between circularly polarized bremsstrahlung photons produced by a longitudinally polarized electron beam striking a copper radiator and the spin-polarized electrons orbiting the iron atoms of an analyzing magnet. This so-called Compton transmission polarimeter is compact and relatively inexpensive compared to Mott-scattering polarimeters because no spin manipulator is required. This work presents the design of the radiator, analyzing magnet, photon detector assembly, and data acquisition system of the Compton transmission polarimeter as well as beam commissioning results performed at the Upgraded Injector Test Facility at Jefferson Lab.