  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.
