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The ultrafast and ultracold electron source, based on laser cooling and trapping of atomic gas and its subsequent near-threshold two-step photoionization, is capable of generating electron bunches with a high transverse brightness at energies of roughly 10 keV. This paper investigates the possibility of increasing the range of applications of this source by accelerating the bunch using radio frequency electromagnetic fields. Bunch energies up to 35 keV are measured by analyzing the diffraction patterns generated from a mono-crystalline gold sample. It is found that the normalized transverse emittance is largely preserved during acceleration.