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We report on direct, real-space imaging of the stray magnetic field above a micro-scale disc of a thin film of the high-temperature superconductor YBa₂Cu₃O_{7&#8722;<i>&#948;</i>} (YBCO) using scanning single spin magnetometry. Our experiments yield a direct measurement of the sample&#8217;s London penetration depth and allow for a quantitative reconstruction of the supercurrents flowing in the sample as a result of Meissner screening. These results show the potential of scanning single spin magnetometry for studies of the nanoscale magnetic properties of thin-film superconductors, which could be readily extended to elevated temperatures or magnetic fields.