The Dalitz decay eta -> e^+ e^- gamma has been measured in the gamma p -> eta p reaction with the Crystal Ball and TAPS multiphoton spectrometers, together with the photon tagging facility at the Mainz Microtron MAMI. The experimental statistic used in this work is one order of magnitude greater than in any previous measurement of eta -> e^+ e^- gamma. The value obtained for the slope parameter 1/Lambda^2 of the eta transition form factor, 1/Lambda^2 = (1.95 +/- 0.15_stat +/- 0.10_syst) [1/GeV^2], is in good agreement with recent measurements conducted in eta -> e^+ e^- gamma and eta -> mu^+ mu^- gamma decays, as well as with recent form-factor calculations. The uncertainty obtained in the value of 1/Lambda^2 is lower compared to results from previous measurements of the eta -> e^+ e^- gamma decay.
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