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An X-ray photoelectric polarimeter based on the Gas Pixel Detector has been proposed to be included in many upcoming space missions to fill the gap of about 30 years from the first (and to date only) positive measurement of polarized X-ray emission from an astrophysical source. The estimated sensitivity of the current prototype peaks at an energy of about 3 keV, but the lack of readily available polarized sources in this energy range has prevented the measurement of detector polarimetric performances. In this paper we present the measurement of the Gas Pixel Detector polarimetric sensitivity at energies of a few keV and the new, light, compact and transportable polarized source that was devised and built to this aim. Polarized photons are produced, from unpolarized radiation generated with an X-ray tube, by means of Bragg diffraction at nearly 45 degrees. The employment of mosaic graphite and flat aluminum crystals allow the production of nearly completely polarized photons at 2.6, 3.7 and 5.2 keV from the diffraction of unpolarized continuum or line emission. The measured modulation factor of the Gas Pixel Detector at these energies is in good agreement with the estimates derived from a Monte Carlo software, which was up to now employed for driving the development of the instrument and for estimating its low energy sensitivity. In this paper we present the excellent polarimetric performance of the Gas Pixel Detector at energies where the peak sensitivity is expected. These measurements not only support our previous claims of high sensitivity but confirm the feasibility of astrophysical X-ray photoelectric polarimetry., Comment: 15 pages, 12 figures. Accepted for publication in NIMA