We report on multifrequency observations performed during 2012 December–2013 August of the first narrow-line Seyfert 1 galaxy detected in γ-rays, PMN J0948+0022 (z = 0.5846). A γ-ray flare was observed by the Large Area Telescope on board Fermi during 2012 December–2013 January, reaching a daily peak flux in the 0.1–100 GeV energy range of (155 ± 31) × 10⁻⁸ ph cm⁻² s⁻¹ on 2013 January 1, corresponding to an apparent isotropic luminosity of ∼1.5 × 10⁴⁸ erg s⁻¹. The γ-ray flaring period triggered Swift and Very Energetic Radiation Imaging Telescope Array System (VERITAS) observations in addition to radio and optical monitoring by Owens Valley Radio Observatory, Monitoring Of Jets in Active galactic nuclei with VLBA Experiments, and Catalina Real-time Transient Survey. A strong flare was observed in optical, UV, and X-rays on 2012 December 30, quasi-simultaneously to the γ-ray flare, reaching a record flux for this source from optical to γ-rays. VERITAS observations at very high energy (E > 100 GeV) during 2013 January 6–17 resulted in an upper limit of F_>0.2 TeV < 4.0 × 10⁻¹² ph cm⁻² s⁻¹. We compared the spectral energy distribution (SED) of the flaring state in 2013 January with that of an intermediate state observed in 2011. The two SEDs, modelled as synchrotron emission and an external Compton scattering of seed photons from a dust torus, can be modelled by changing both the electron distribution parameters and the magnetic field.