We report on the design and performance of the LUX-ZEPLIN (LZ) “Screener”, a small liquid scintillator detector consisting of ≈23 kg of LAB-based gadolinium-loaded liquid scintillator (GdLS) to be used in the LZ Outer Detector. The Outer Detector will be filled with 17.3 tonnes of GdLS and will surround the central liquid xenon time projection chamber of LZ. Its primary function will be to tag neutron events in the liquid xenon which could mimic a WIMP dark matter signal. To meet the deadtime requirements for the Outer Detector, the radioimpurity levels in the GdLS must be kept below ≲0.07 mBq/kg. This background level corresponds to a rate of ≈50Hz above an energy threshold of 100 keV. The Screener was operated in the ultra-low-background environment of the former LUX water shield in the Davis Laboratory at the Sanford Underground Research Facility for radioassay of the GdLS. Careful selection of detector materials and use of ultra-low-background PMTs allows the measurement of a variety of radioimpurities. The 14C/12C ratio in the scintillator is measured to be (2.83±0.06(stat.)±0.01(sys.))×10−17. Use of pulse shape discrimination allows the concentration of isotopes throughout the 238U, 235U, and 232Th chains to be measured by fitting the collected spectra from α and β events. We find that equilibrium is broken in the 238U and 232Th chains and that a significant portion of the contamination in the GdLS results from decays in the 227Ac subchain of the 235U series. Predictions for the singles rate in the Outer Detector are presented. The rate from radioimpurities above 100 keV in the GdLS is estimated to be 97.9±6.4Hz, with 65.5±1.9Hz resulting from α-decays.