We report the first measurement of the photoproduction cross section of the $\phi$ meson in its neutral decay mode in the reaction $\gamma p \to p\phi(K_SK_L)$. The experiment was performed with a tagged photon beam of energy $1.6 \le E_\gamma \le 3.6$ GeV incident on a liquid hydrogen target of the CLAS spectrometer at the Thomas Jefferson National Accelerator Facility. The $p \phi$ final state is identified via reconstruction of $K_S$ in the invariant mass of two oppositely charged pions and by requiring the missing particle in the reaction $\gamma p \to p K_S X$ to be $K_L$. The presented results significantly enlarge the existing data on $\phi$-photoproduction. These data, combined with the data from the charged decay mode, will help to constrain different mechanisms of $\phi$ photoproduction.