8 pages, 4 figures. Accepted for publication in ApJ Letter. --Corresponding authors: Masaaki Hayashida (mahaya_at_icrr.u-tokyo.ac.jp), Greg Madejski (madejski_at_slac.stanford.edu), and Krzysztof Nalewajko (knalew_at_camk.edu.pl) ; International audience ; On 2015 June 16, Fermi-LAT observed a giant outburst from the flat spectrum radio quasar 3C 279 with a peak $>100$ MeV flux of $∼3.6\times10^{-5}\;{\rm photons}\;{\rm cm}^{-2}\;{\rm s}^{-1}$ averaged over orbital period intervals. It is the historically highest $γ$-ray flux observed from the source including past EGRET observations, with the $γ$-ray isotropic luminosity reaching $∼10^{49}\;{\rm erg}\;{\rm s}^{-1}$. During the outburst, the Fermi spacecraft, which has an orbital period of 95.4 min, was operated in a special pointing mode to optimize the exposure for 3C 279. For the first time, significant flux variability at sub-orbital timescales was found in blazar observations by Fermi-LAT. The source flux variability was resolved down to 2-min binned timescales, with flux doubling times less than 5 min. The observed minute-scale variability suggests a very compact emission region at hundreds of Schwarzschild radii from the central engine in conical jet models. A minimum bulk jet Lorentz factor ($Γ$) of 35 is necessary to avoid both internal $γ$-ray absorption and super-Eddington jet power. In the standard external-radiation-Comptonization scenario, $Γ$ should be at least 50 to avoid overproducing the synchrotron-self-Compton component. However, this predicts extremely low magnetization ($∼5\times10^{-4}$). Equipartition requires $Γ$ as high as 120, unless the emitting region is a small fraction of the dissipation region. Alternatively, we consider $γ$ rays originating as synchrotron radiation of $γ_{\rm e}∼1.6\times10^6$ electrons, in magnetic field $B∼1.3$ kG, accelerated by strong electric fields $E∼ B$ in the process of magnetoluminescence. At such short distance scales, one cannot immediately exclude production of $γ$ rays in hadronic processes.