The Gas Pixel Detector belongs to the very limited class of gas detectors optimized for the measurement of X-ray polarization in the emission of astrophysical sources. The choice of the mixture in which X-ray photons are absorbed and photoelectrons propagate, deeply affects both the energy range of the instrument and its performance in terms of gain, track dimension and ultimately, polarimetric sensitivity. Here we present the characterization of the Gas Pixel Detector with a 1 cm thick cell filled with dimethyl ether (DME) at 0.79 atm, selected among other mixtures for the very low diffusion coefficient. Almost completely polarized and monochromatic photons were produced at the calibration facility built at INAF/IASF-Rome exploiting Bragg diffraction at nearly 45 degrees. For the first time ever, we measured the modulation factor and the spectral capabilities of the instrument at energies as low as 2.0 key, but also at 2.6, 3.7, 4.0, 5.2 and 7.8 key. These measurements cover almost completely the energy range of the instrument and allows to compare the sensitivity achieved with that of the standard mixture, composed of helium and DME. (C) 2010 Elsevier B.V. All rights reserved.