ABSTRACT We report the results of a multiband observing campaign on the famous blazar 3C 279 conducted during a phase of increased activity from 2013 December to 2014 April, including first observations of it with NuSTAR. The γ-ray emission of the source measured by Fermi-LAT showed multiple distinct flares reaching the highest flux level measured in this object since the beginning of the Fermi mission, with of 10−5 photons cm−2 s−1, and with a flux-doubling time scale as short as 2 hr. The γ-ray spectrum during one of the flares was very hard, with an index of , which is rarely seen in flat-spectrum radio quasars. The lack of concurrent optical variability implies a very high Compton dominance parameter . Two 1 day NuSTAR observations with accompanying Swift pointings were separated by 2 weeks, probing different levels of source activity. While the 0.5−70 keV X-ray spectrum obtained during the first pointing, and fitted jointly with Swift-XRT is well-described by a simple power law, the second joint observation showed an unusual spectral structure: the spectrum softens by at ∼4 keV. Modeling the broadband spectral energy distribution during this flare with the standard synchrotron plus inverse-Compton model requires: (1) the location of the γ-ray emitting region is comparable with the broad-line region radius, (2) a very hard electron energy distribution index , (3) total jet power significantly exceeding the accretion-disk luminosity , and (4) extremely low jet magnetization with . We also find that single-zone models that match the observed γ-ray and optical spectra cannot satisfactorily explain the production of X-ray emission.