  The mysterious pseudogap phase of cuprate superconductors ends at a critical
hole doping level p* but the nature of the ground state below p* is still
debated. Here, we show that the genuine nature of the magnetic ground state in
La2-xSrxCuO4 is hidden by competing effects from superconductivity: applying
intense magnetic fields to quench superconductivity, we uncover the presence of
glassy antiferromagnetic order up to the pseudogap boundary p* ~ 0.19, and not
above. There is thus a quantum phase transition at p*, which is likely to
underlie highfield observations of a fundamental change in electronic
properties across p*. Furthermore, the continuous presence of quasi-static
moments from the insulator up to p* suggests that the physics of the doped Mott
insulator is relevant through the entire pseudogap regime and might be more
fundamentally driving the transition at p* than just spin or charge ordering.
