Very high energy (VHE, E > 100 GeV)gamma-ray flaring activity of the high-frequency peaked BL Lac object PG 1553 + 113 has been detected by the H.E.S.S. telescopes. The flux of the source increased by a factor of 3 during the nights of 2012 April 26 and 27 with respect to the archival measurements with a hint of intra-night variability. No counterpart of this event has been detected in the Fermi-Large Area Telescope data. This pattern is consistent with VHE gamma(-)ray flaring being caused by the injection of ultrarelativistic particles, emitting.-rays at the highest energies. The dataset offers a unique opportunity to constrain the redshift of this source at z = 0.49 +/- 0.04 using a novel method based on Bayesian statistics. The indication of intra-night variability is used to introduce a novel method to probe for a possible Lorentz invariance violation (LIV), and to set limits on the energy scale at which Quantum Gravity (QG) effects causing LIV may arise. For the subluminal case, the derived limits are E-QG,E- 1 > 4.10 x 10(17) GeV and E-QG,E- 2 > 2.10 x 10(10) GeV for linear and quadratic LIV effects, respectively.