Symmetry-breaking quantum phase transitions play a key role in several condensed matter, cosmology and nuclear physics theoretical models1-3. Its observation in real systems is often hampered by finite temperatures and limited control of the system parameters. In this work we report for the first time the experimental observation of the full quantum phase diagram across a transition where the spatial parity symmetry is broken. Our system is made of an ultra-cold gas with tunable attractive interactions trapped in a spatially symmetric double-well potential. At a critical value of the interaction strength, we observe a continuous quantum phase transition where the gas spontaneously localizes in one well or the other, thus breaking the underlying symmetry of the system. Furthermore, we show the robustness of the asymmetric state against controlled energy mismatch between the two wells. This is the result of hysteresis associated with an additional discontinuous quantum phase transition that we fully characterize. Our results pave the way to the study of quantum critical phenomena at finite temperature4, the investigation of macroscopic quantum tunneling of the order parameter in the hysteretic regime and the production of strongly quantum entangled states at critical points5.