Reaching light intensities above 1025 W/cm2 and up to the Schwinger limit of the order of 1029 W/cm2 would enable testing fundamental predictions of quantum electrodynamics. A promising - yet challenging - approach to achieve such extreme fields consists in reflecting a high-power femtosecond laser pulse off a curved relativistic mirror. This enhances the intensity of the reflected beam by simultaneously compressing it in time down to the attosecond range, and focusing it to sub-micrometre focal spots. Here we show that such curved relativistic mirrors can be produced when an ultra-intense laser pulse ionizes a solid target and creates a dense plasma that specularly reflects the incident light. This is evidenced by measuring the temporal and spatial effects induced on the reflected beam by this so-called 'plasma mirror'. The all-optical measurement technique demonstrated here will be instrumental for the use of relativistic plasma mirrors with the upcoming generation of Petawatt lasers that recently reached intensities of 5 × 1022 W/cm2, and therefore constitutes a viable experimental path to the Schwinger limit.