The hitherto unexplored two-photon doubly-excited states [Ne$^{*}$($2p^{-1}3s$)]$_{2}$ were experimentally identified using the seeded, fully coherent, intense extreme ultraviolet free-electron laser FERMI. These states undergo ultrafast interatomic Coulombic decay (ICD) which predominantly produces singly-ionized dimers. In order to obtain the rate of ICD, the resulting yield of Ne$_{2}^{+}$ ions was recorded as a function of delay between the XUV pump and UV probe laser pulses. The extracted lifetimes of the long-lived doubly-excited states, 390 (-130 / +450} fs, and of the short-lived ones, less than 150~fs, are in good agreement with emph{ab initio} quantum mechanical calculations.