The quark structure of the $f_2(1270)$ meson has, for many years, been assumed to be a pure quark-antiquark ($qbar{q}$) 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 $rho$-mesons. Such a state would be expected to decay strongly to final states with charged pions, due to the dominant decay $rho to pi^+ pi^-$, whereas decay to two neutral pions would likely be suppressed. Here, we measure for the first time the reaction $gamma p to pi^0 pi^0 p$, using the CLAS detector at Jefferson Lab for incident beam energies between 3.6-5.4~GeV. Differential cross sections, $dsigma / dt$, for $f_2(1270)$ photoproduction are extracted with good precision, due to low backgrounds, and are compared with theoretical calculations.