We report on the results of the first measurement of exclusive $f_0(980)$ meson photoproduction on protons for $E_gamma=3.0 - 3.8$ GeV and $-t = 0.4-1.0$ GeV$^2$. Data were collected with the CLAS detector at the Thomas Jefferson National Accelerator Facility. The resonance was detected via its decay in the $pi^+ pi^-$ channel by performing a partial wave analysis of the reaction $gamma p to p pi^+ pi^-$. Clear evidence of the $f_0(980)$ meson was found in the interference between $P$ and $S$ waves at $M_{pi^+ pi^-}sim 1$ GeV. The $S$-wave differential cross section integrated in the mass range of the $f_0(980)$ was found to be a factor of 50 smaller than the cross section for the $rho$ meson. This is the first time the $f_0(980)$ meson has been measured in a photoproduction experiment.