An earlier measurement on the 4+ to 2+ radiative transition in 8Be provided the first electromagnetic signature of its dumbbell-like shape. However, the large uncertainty in the measured cross section does not allow a stringent test of nuclear structure models. This Letter reports a more elaborate and precise measurement for this transition, via the radiative capture in the 4He + 4He reaction, improving the accuracy by about a factor of 3. Ab initio calculations of the radiative transition strength with improved three-nucleon forces are also presented. The experimental results are compared with the predictions of the alpha cluster model and ab initio calculations.