We demonstrate plasma-mirror frequency-resolved optical gating (PM-FROG) using a liquid-sheet jet of water and characterize a waveform of a high-repetition laser pulse (1 kHz) in the ultraviolet (UV) region. The measured PM-FROG trace is reconstructed by the least-squares generalized projections algorithm. The retrieved UV spectral phase is consistent with that by self-diffraction FROG. The complex reflection coefficient of a plasma mirror is discussed in terms of the free electron density dependence. The PM-FROG technique is applicable to not only the pulse characterization, but also the investigation of ionization dynamics of a liquid.