The quark structure of the f_{2}(1270) meson has, for many years, been assumed to be a pure quark-antiquark (qq[over ¯]) resonance with quantum numbers J^{PC}=2^{++}. Recently, it was proposed that the f_{2}(1270) is a molecular state made from the attractive interaction of two ρ mesons. Such a state would be expected to decay strongly to final states with charged pions due to the dominant decay ρ→π^{+}π^{-}, whereas decay to two neutral pions would likely be suppressed. Here, we measure for the first time the reaction γp→π^{0}π^{0}p, using the CEBAF Large Acceptance Spectrometer detector at Jefferson Lab for incident beam energies between 3.6 and 5.4 GeV. Differential cross sections, dσ/dt, for f_{2}(1270) photoproduction are extracted with good precision due to low backgrounds and are compared to theoretical calculations.