Josephson plasma waves are linear electromagnetic modes that propagate along the planes of cuprate superconductors, sustained by interlayer tunnelling supercurrents. For strong electromagnetic fields, as the supercurrents approach the critical value, the electrodynamics become highly nonlinear. Josephson plasma solitons (JPSs) are breather excitations predicted in this regime, bound vortex-antivortex pairs that propagate coherently without dispersion. We experimentally demonstrate the excitation of a JPS in La1.84Sr0.16CuO4, using intense narrowband radiation from an infrared free-electron laser tuned to the 2-THz Josephson plasma resonance. The JPS becomes observable as it causes a transparency window in the opaque spectral region immediately below the plasma resonance. Optical control of magnetic-flux-carrying solitons may lead to new applications in terahertz-frequency plasmonics, in information storage and transport and in the manipulation of high-T-c superconductivity.