  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
(pPb) 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 nb$^{-1}$. The two J/$\psi$ mesons are reconstructed in
their $\mu^+\mu^-$ decay channels with transverse momenta $p_\mathrm{T}$ $\gt$
6.5 GeV and rapidity $\lvert y \rvert$ $\lt$ 2.4. Events where one of the
J/$\psi$ mesons is reconstructed in the dielectron channel are also considered
in the search. The pPb $\to$ 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 $\sigma$(pPb$\to$
J/$\psi$J/$\psi$+X)= 22.0 $\pm$ 8.9 (stat) $\pm$ 1.5 (syst) 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
$\sigma^{\mathrm{pPb}\to\mathrm{J}/\psi\mathrm{J}/\psi+\mathrm{X}}_\text{SPS}$
= 16.5 $\pm$ 10.8 (stat) $\pm$ 0.1 (syst) nb and $\sigma^{\mathrm{pPb}\to
\mathrm{J}/\psi\mathrm{J}/\psi+\mathrm{X}}_\text{DPS}$ = 5.4 $\pm$ 6.2 (stat)
$\pm$ 0.4 (syst) 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
$\sigma_\text{eff}$ $\gt$ 1.0 mb at 95% confidence level.
