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We have developed a new compact high average current dc superconducting rf (DC-SRF) photoinjector. A 3.5-cell superconducting cavity with a mid-beta structure is the key component of the DC-SRF photoinjector. Herein, we present the new design, fabrication, and vertical test results of the 3.5-cell large grain niobium superconducting cavity. After preliminary surface preparation and room temperature rf measurements, the cavity was sent to Jefferson Lab for a vertical cold test. The gradient of the cavity reaches 23.5  MV/m, and the Q_{0} value is higher than 1.2×10^{10}. The high-quality 3.5-cell cavity will guarantee the good performance of the DC-SRF photoinjector.