Production of prompt J/psi meson pairs in proton-proton collisions at root s = 7 TeV is measured with the CMS experiment at the LHC in a data sample corresponding to an integrated luminosity of about 4.7 fb(-1). The two J/psi mesons are fully reconstructed via their decays into mu(+)mu(-) pairs. This observation provides for the first time access to the high-transverse-momentum region of J/psi pair production where model predictions are not yet established. The total and differential cross sections are measured in a phase space defined by the individual J/psi transverse momentum (p(T)(J/psi)) and rapidity (vertical bar y(J/psi)vertical bar): vertical bar y(J/psi)vertical bar < 1.2 for p(T)(J/psi) > 6.5 GeV/c; 1.2 < vertical bar y(J/psi)vertical bar < 1.43 for a pT threshold that scales linearly with vertical bar y(J/psi)vertical bar from 6.5 to 4.5 GeV/c; and 1.43 < vertical bar y(J/psi)vertical bar < 2.2 for p(T)(J/psi) > 4.5 GeV/c. The total cross section, assuming unpolarized prompt J/psi pair production is 1.49 +/- 0.07 (stat) +/- 0.13 (syst) nb. Different assumptions about the J/psi polarization imply modifications to the cross section ranging from -31% to +27%.