Magnetic reconnection is thought to drive a wide variety of dynamic phenomena in the solar atmosphere. Yet, the detailed physical mechanisms driving reconnection are difficult to discern in the remote sensing observations that are used to study the solar atmosphere. In this Letter, we exploit the high-resolution instruments Interface Region Imaging Spectrograph and the new CHROMIS Fabry Perot instrument at the Swedish 1-m Solar Telescope (SST) to identify the intermittency of magnetic reconnection and its association with the formation of plasmoids in socalled UV bursts in the low solar atmosphere. The Si IV 1403 angstrom UV burst spectra from the transition region show evidence of highly broadened line profiles with often non-Gaussian and triangular shapes, in addition to signatures of bidirectional flows. Such profiles had previously been linked, in idealized numerical simulations, to magnetic reconnection driven by the plasmoid instability. Simultaneous CHROMIS images in the chromospheric Ca 11 K 3934 angstrom line now provide compelling evidence for the presence of plasmoids by revealing highly dynamic and rapidly moving brightenings that are smaller than 0.12 and that evolve on timescales of the order of seconds. Our interpretation of the observations is supported by detailed comparisons with synthetic observables from advanced numerical simulations of magnetic reconnection and associated plasmoids in the chromosphere. Our results highlight how subarcsecond imaging spectroscopy sensitive to a wide range of temperatures combined with advanced numerical simulations that are realistic enough to compare with observations can directly reveal the small-scale physical processes that drive the wide range of phenomena in the solar atmosphere.