We present the observation of doubly-produced $J/\psi$ mesons with the D0 detector at Fermilab in $p\bar{p}$ collisions at $\sqrt{s}=1.96$ TeV. The production cross section for both singly and doubly-produced $J/\psi$ mesons is measured using a sample with an integrated luminosity of 8.1fb$^{-1}$. For the first time, the double $J/\psi$ production cross section is separated into contributions due to single and double parton scatterings. Using these measurements, we determine the effective cross section $\sigma_{eff}$, a parameter characterizing an effective spatial area of the parton-parton interactions and related to the parton spatial density inside the nucleon.