The first observation of the concurrent production of two$J/\psi$mesons in proton-nucleus collisions is presented. The analysis is based on a proton-lead ($p\mathrm{Pb}$) data sample recorded at a nucleon-nucleon center-of-mass energy of 8.16 TeV by the CMS experiment at the CERN LHC and corresponding to an integrated luminosity of$174.6{\mathrm{nb}}^{-1}$. The two$J/\psi$mesons are reconstructed in their${\mu }^{+}{\mu }^{-}$decay channels with transverse momenta$p_{\mathrm{T}}>6.5\mathrm{GeV}$and rapidity$|y|<2.4$. Events where one of the$J/\psi$mesons is reconstructed in the dielectron channel are also considered in the search. The$p\mathrm{Pb}\to J/\psi J/\psi +X$process is observed with a significance of 5.3 standard deviations. The measured inclusive fiducial cross section, using the four-muon channel alone, is$\sigma (p\mathrm{Pb}\to J/\psi J/\psi +X)=22.0\pm 8.9(\mathrm{stat})\pm 1.5(\mathrm{syst})\mathrm{nb}$. A fit of the data to the expected rapidity separation for pairs of$J/\psi$mesons produced in single (SPS) and double (DPS) parton scatterings yields${\sigma }_{\mathrm{SPS}}^{p\mathrm{Pb}\to J/\psi J/\psi +X}=16.5\pm 10.8(\mathrm{stat})\pm 0.1(\mathrm{syst})\mathrm{nb}$and${\sigma }_{\mathrm{DPS}}^{p\mathrm{Pb}\to J/\psi J/\psi +X}=5.4\pm 6.2(\mathrm{stat})\pm 0.4(\mathrm{syst})\mathrm{nb}$, respectively. This latter result can be transformed into a lower bound on the effective DPS cross section, closely related to the squared average interparton transverse separation in the collision, of${\sigma }_{\mathrm{eff}}>1.0\mathrm{mb}$at 95% confidence level.