It is a long-standing nontrivial prediction of quantum electrodynamics that its vacuum is unstable in the background of a static, spatially uniform electric field and, in principle, sparks with spontaneous emission of electron-positron pairs. However, an experimental verification of this prediction seems out of reach because a sizeable rate for spontaneous pair production requires an extraordinarily strong electric field strength |E| of order the Schwinger critical field, E_c=m_e²/e≃1.3×10¹⁸  V/m, where me is the electron mass and e is its charge. Here, we show that the measurement of the rate of pair production due to the decays of high-energy bremsstrahlung photons in a high-intensity laser field allows for the experimental determination of the Schwinger critical field and, thus, the boiling point of the vacuum of quantum electrodynamics.